Short Answer
In the SWU, there are two types of deflector shields (ray shields and particle shields), and two types of conventional projectile weapons (energy weapons and physical weapons). Ray shields block energy weapons and particle shields block physical weapons (and possibly energy weapons as well). In-universe, an example of an energy weapon is a blaster bolt or a turbolaser, whereas proton weapons are an example of a physical weapon.
All of the examples the OP is listing involved ray shields, not particle shields. The Acclamator-class ships were known to be vulnerable to ground fire and were never stated to have full particle shields, the LAATs are small ships and small ships almost always use only ray shields (if even that), and the thermal exhaust ports on the First Death Star were ray shielded, not particle shielded.
As physical weapons, all proton weapons can penetrate ray shields, but not particle shields. In the instances listed by the OP, the proton weapons were either bypassing ray shields or no shields at all, which is precisely what they are desgined to do, not because of the ‘proton’ part, but simply because they are physical weapons (of the proton variety) and thus can penetrate ray shields. This is discussed in greater depth below in the Long Answer.
Out of universe the distinction between energy weapons and physical weapons, as well as the effective difference between a ray shield and a particle shield (based on how they probably work), is not entirely clear, as I discuss, again, below in the Long Answer.
Long Answer
Introduction
In-universe, as Null brings up in his comment, there is a distinction, between ray shields and particle shields, and that is the key to answering this question. There are two types of deflector shield, ray shields and particle shields:
Ray shields are:
designed to block radiation (or at least some of it, communications and sensors and visible light (you can see the stars from the windows) could still get through, as touched on here)
designed to block ‘energy’ weapons such as blasters or laser cannons
often the only shields used on smaller craft such as starfighters
used to cover the thermal exhaust ports on the Death Star since particle shields would apparently prevent it from venting (the exhaust ports vent heat (radiation) and gases, ray shields keep out ions and radiation, but if the gases are what carry the heat via convection then it makes sense, there wouldn’t be much conduction in the near-vacuum of space, and I don’t know how much of the heat waste would be radiation)
Particle shields:
are designed to block ‘physical’ weapons such as artillery and missiles
are also used to block asteroids, meteors and space debris
are often used by larger ships, also in conjunction with ray shields
have to be disabled if a ship wants to fire a ‘physical’ weapon through them
The physics of it
The science/physics in Star Wars when it comes to weapons and shields is not entirely consistent or correct. My physics isn’t the best in the world, but I believe the below to be generally accurate.
The Weapons
The term ‘energy weapon’ is misleading, since it encompasses two different kinds of weapons, lasers and particle beam weapons. Lasers are used almost never used (and discussed in the top answer here) in the Star Wars universe. All of the other ‘energy’ weapons are actually particle beam weapons on one sort or another. Particle beam weapons are kinetic weapons, not energy weapons, they don’t rely on the effects of radiation talked about here, they rely on the kinetic impact of zillions of tiny charged particles/ions accelerated to just below the speed of light slamming into things
Laser cannons, turbolasers and superlasers (such as the Death Star superlaser, which I talk about here) are all particle beam weapons, not lasers
Ion cannons are particle beam weapons, which have the special property of being able to disable a ship’s electronics and power systems
Blasters are particle beam weapons as well, not lasers (as discussed here), they ionise ‘blaster gas’ (either with lasers or electric fields) into compressed packages or ‘bolts’ of charged particles, and then shoot those (somehow) coherent bolts at a very fast speed
Lightsabers are some sort of magnetic plasma bottle, not lasers/light, as is discussed at the top of their Wookiee article:
The weapon consisted of a blade of pure plasma emitted from the hilt
and suspended in a force containment field. The field contained the
immense heat of the plasma, protecting the wielder, and allowed the
blade to keep its shape.
In the Star Wars Universe, a proton torpedo is a physical missile that carried a warhead which somehow scattered a payload of protons to tremendous destructive effect
Proton cannons shoot proton artillery which presumably operate along the same lines as a proton torpedo
The Shields
The Wookieepedia article on Deflector Shields describes them thusly:
Deflector shields worked in a layered defense fashion: A volumetric
field effect extended out from the surface of the shield projector,
attempting to reduce the coherency of any beam attacks and deflect
physical objects. The shield itself behaved in a manner similar to
that of a thermally conductive material—energy applied was quickly
diffused and re-radiated back into the environment, but the shield
itself could also absorb some of the energy. The absorbed energy was
shunted into heat sinks, and re-radiated at a lesser rate by the
shield and neutrino radiators. The use of deflector shields on
starfighters and other small craft, made them capable of hypersonic
speeds in planetary atmospheres.
This explanation conflates ray shields and particle shields. All physical particles sometimes display wave-like properties, such as coherence, but I am presuming they are only talking about 'beams’ of light, not protons in superposition. By ‘absorb energy’ I am guessing they mean absorbing kinetic energy from impact, and then ‘re-radiating’ has something to do with the heat generated by absorption/deflection process, since all of the weapons it blocks are actually kinetic weapons, not lasers ‘heating up’ the shield.
In real life physics I only know of two main ways of blocking a laser:
the first is with a mirror to reflect it (if the laser isn’t too strong),
the second is if you can generate a plasma window with enough plasma in just the right way, it can absorb and/or deflect at least some of the radiation (as discussed in this article). Very powerful real-world lasers are used to ionise gases so there is at least some interaction there, as discussed here, which also means that at least some of the laser’s energy can be dissipated passing through gases. Electrically neutral gases would absorb some of the radiation as ionization energy (and refract some of it), electrically charged gases (plasma is usually ionised gas) could deflect it
Blocking charged particles, i.e. particle beam weapons, can be accomplished with very powerful magnetic fields (by deflecting the particles, like how particle accelerators move charged particles around), or with the above plasma window design (by absorbing and/or deflecting them) since the magnetic fields required to hold the plasma in place can deflect the charged particles which can also be absorbed by/deflected off of the plasma itself.
I cannot find any explanation in the Star Wars universe for how a 'particle’ shield deflects/blocks physical weapons such as artillery or missiles. I can only surmise that, again, some combination of very powerful magnetic fields (possibly to ionise the 'tip’ of the warhead and then deflect it or at least slow it down so that it impacts the plasma window) and/or plasma windows (to physically impede and impact the projectile) are used. It is doubtful that particle shields are only designed to stop the charged particle payload in weapons, as opposed to the physical weapon itself, since they are also used against space debris and asteroids/meteors. Meteors ionise the atmosphere when they enter it, and comets are ionised by solar radiation, but there is no evidence that that is what the particle shields are deflecting, or that those ions exist within enough abundance to be used to stop the entire mass of the debris.
The conclusion therefore is that if you were using real world physics, the methods for blocking lasers, particle beam weapons, and physical projectiles (with or without ion payloads) should all be identical, so there should not even be a distinction between a ray shield and a particle shield, unless one is just weaker or differently modulated than the other. In-universe, however, this conclusion is difficult to sustain due to three main inconsistencies:
Ray shields are shown letting electrically neutral gases pass through them (Death Star thermal exhaust ports), yet somehow they do not ionise these gases on the way out and thus turn them into the sort of thing (particle beams) that they block.
Ray shields do not allow humans to pass through them (Anakin and Obi-wan get trapped in one at the beginning of ROTS) without getting vaporised, yet they are not considered to be effective enough in blocking out space debris to the extent that particle shields are required for that
Presuming that the shields do operate along the principles of a plasma window, an immediate inconsistency is that proton torpedos, like those used to destroy the first Death Star, still carry charged particles/ions that would have, at some point, to pass through the ray shield’s magnetic field/plasma window, so it is unclear why a proton torpedo would get through a ray shield but a particle beam made up of protons would not, unless a) it has enough momentum to do so, or b) it may be possible that the proton warhead is shielded or the overall weapon is electrically neutral, but that is never stated as far as I can find.
Based on all of this, I cannot find a satisfactory (non-handwavy) explanation for why a proton weapon can get through a ray shield but not a particle shield. The only plausible method of blocking lasers and charged particle beams is the same method for blocking proton weapons - some version of a plasma window - yet the in-universe evidence indicates that they are not operating on the same principles.
The only alternative explanation I can think of is that particle shields are magnetic fields only, with no plasma, and thus let radiation in but somehow deflect/vaporise incoming physical objects, and to do that, the magnetic fields would have to be ridiculously powerful, if it’s even possible. However, this still would not explain why you would then not use these magnetic fields to just deflect charged particle beam weapons, in which case, why bother having ray shields for ship-defense (or anything else, the particle shields would be just as good at containing captured objects or protecting droidekas et cetera), unless there is some engineering or power-consumption distinction, which, again, does not appear in the available materials.
OP’s Question
In the examples the OP provides:
The Acclamator-class assault ship should by all accounts possess both ray and particle shielding since it was a large military ship, but it does not say so definitively anywhere. Its primary purpose was to carry and deploy troops and assault craft. These ships may only carry ray shielding since their primary purpose was to deploy troops from orbit and then act as an orbital command center, or to deploy them from the surface in favorable conditions. The Star Wars Saga (Legends) Starships of the Galaxy (pg. 56) states that they were known to be vulnerable to ground fire, and the Complete Star Wars Encyclopedia states (on pg. 8) that they were capable of surviving space battles, so it is possible that they were only given ray shields to survive the 'laser/energy’ weapons of opponent ships. They were also known to be exceptionally fast, so perhaps instead of using energy for particle shields, they preferred it for the engines, but that is entirely speculation. If they are only protected by ray shielding, then that would explain (in-universe) why they could be harmed by physical proton cannon artillery.
Smaller ships, as mentioned at the top of this answer, typically only carry ray shielding, and this could be true of the LAAT’s (which may not have had any shielding at all), so again, in-universe, physical proton cannon artillery could hit them.
The Death Star thermal exhaust ports are ray shielded, and not particle shielded, so physical proton torpedos could get through them.
The conclusion, in-universe, is that:
the term 'proton weapon’ refers to a physical weapon like artillery or a missile, which carries a warhead of ions/charged particles, which may somehow be electrically neutral overall, and/or whose payload may somehow be magnetically shielded, and which can penetrate ray shields but not particle shields
the words laser and ion weapon actually refer to (charged/)particle beam weapons, which are blocked by ray shields and presumably by particle shields as well since particle beams are still physical objects (ions) and would be affected by the magnetic fields/plasma window design of the particle shield, if it works that way
The OP states:
There are at least two instances where proton weapons have penetrated
military shields specifically because they are proton weapons
From the available evidence, the proton weapons are not bypassing particle shields which are designed to stop them, they are bypassing ray shields which are not capable of stopping them, so this is not evidence of a special property of proton weapons per say, i.e. these instances are not evidence that proton weapons, as opposed to some other kind of physical weapon, can especially bypass particle shields, nor are they examples of proton weapons being the only physical weapons to bypass ray shields, all that we are seeing is proton weapons bypassing ray shields, which is what they are designed to do. Proton weapons cannot penetrate all shields, just ray shields, and they can do this because they are physical weapons (as opposed to energy weapons, which ray shields block).
N.B.1. Out of universe this conclusion still doesn’t make much sense since protons are ions (the ions of hydrogen) and therefore proton weapons carry ions, which presumably would be blocked by the same mechanism which is used to block lasers and particle beams, and thus would retard the entire physical weapon (or explode it) in a manner similar to how a casing-less collection of ions/charged particles would be blocked, unless, as stated above, proton weapons are either electrically neutral overall or whose payload is somehow magnetically shielded. But that is out of universe …
N.B.2. The general inconsistency of ray shields, as Null linked to in his comment, has been discussed before. There does not appear to be any particularly satisfying explanation for why ray shields would vaporise humans but not artillery shells or space debris. There also is no real explanation for why, when big ships become depressurised, and people/droids get expelled into space, they aren’t vaporised when there is no evidence that the ray shields themselves are necessarily down especially since the shields cover the entire ship, especially the observation decks to protect them from frontal assault. Maybe each time it happens in the movies and TCW the shields just happen to be down. Visually, ray shields are never even shown around the ship, which may indicate that they, unlike particle shields, are only ever turned on when they are actually needed, they are not on by default.
N.B.3. The blue shields seen in various Star Wars media around the openings into space, such as at the top of Acclamator-class ships, is neither a particle shield, nor a ray shield, but some other kind of shield which is discussed here.
N.B.4. Also note that based on the foregoing theory, if it is assumed that particle shields are more powerful versions of ray shields, and thus can defend against particle beam weapons as well as physical ones, a particle beam weapon strong enough can still break through those shields, as evidenced by the Death Star destroying Alderaan (which had a planetary shield, and Admiral Ackbar’s words in ROTJ - youtube clip.
N.B.5. The Gungan shields in TPM don’t seem to be either deflector shields or ray shields, they block blaster bolts (particle beam weapons) but let the droids though, so unless ray shields vaporise humans but not robots, they are’t ray shields, and they aren’t particle shields since particle shields block physical objects from passing through them. This topic is discussed here. The in-universe explanation of them (that they only blocked things with a high-energy input) cannot explain the distinction between ray shields and particle shields since both block high-energy input mechanisms (blaster bolts and proton missiles).
