This answer contains spoilers for multiple episodes.
I would think someone more qualified than me could make a good argument for Picard. However, I will give this a start to mention some of the more unique assignments. There's...
- (surprise! Picard's been a captain of a starship!)
- Assigned by Rear Admiral Noral Satie, Starfleet command, Stardate 41148)
- Arbiter of (Klingon) Succession (Memory Alpha: TnG: Reunion)
- assigned by Klingon Chancellor K'mpec
- Worf's Cha'DIch (Memory Alpha: TnG: Sins of the Father)
- Picard fills in after Commander Kurn is attacked
- This seems to be a very official position in the Klingon judicial system. (It led to Picard being in room, having a rather private conversation with the Klingon Chancellor.)
- Undercover operative (Memory Alpha: Chain of Command)
- Vice Admiral Nechayev who then assigns Jellico as Captain of the Enterprise (so Picard is, quite clearly, no longer in the same position of being active Captain of the Enterprise). Since Picard is no longer Captain of the Enterprise, he is clearly being officially assigned a different position. Picard's role as captain was replaced with this role as undercover operative (assigned by the Federation), until that was effectively replaced by the Cardassians who assigned him a new role...
- "Human" (Memory Alpha: Chain of Command, Part 2)
- No, the word "human" is not just Picard's species. This is different than when many characters call Worf by a reference of "Klingon". When Picard is a prisoner, a Cardassian Interrogator/Torturer tells Picard to expect to be referred to "only as human. You have no other identity." This is an assigned by someone with some local authority.
However, there is one argument that could be made that this isn't clearly an official position. Although prison populations can have some hierarchy that may have offered Picard some changes, there's the compelling case that Picard may have retained his permanently-assigned identity even if his position changed to something else (like an enslavened servant of someone of high official rank).
- Ambassador to Humanity (Memory Alpha: The Best of Both Worlds)
- This is a position assigned to him by the Borg Collective. This official role is basically to let Humanity know the Borg's directives. Locutus of Borg fully complies with the expectations of this position.
- Dead. (Memory Alpha: TNG episode named "Gambit")
This is actually my personal favorite part of the whole answer. However, some people understandably don't consider "dead" to be a position. Well, if "Prisoner of War" is a recognized role/title, then I would suggest that so is "casualty". (So I propose using that term if you find that easier to accept.)
Although this assignment/declaration was based on incorrect information, the Federation officially considered him to be dead for some time, as local evidence indicated he was killed. Starfleet invalidated his command codes, which demonstrates Picard's officially assigned and recognized status was "dead".
Even once we the viewing audience found Picard was acting as a mercenary named Galen, this doesn't mean that Starfleet knew this quite so early. Starfleet still thought him to be dead for some time yet.
Starfleet apparently has not officially updated Picard's status by the end of this two-part episode, which plays a part in the humorous final scene (where Data is seen sending Riker to the brig).
Granted, Riker's role was "Acting Captain" until they got all this sorted out. So that wouldn't fit the criteria being asked for. But Picard's assigned position is different because that assignment wasn't meant to be a temporary thing. This was not a temporary ruse as part of an undercover mission. The intent was an unfortunate expectation of Picard keeping this status in the (very) long term. (This position definitely comes with a change in expected duties.)
- Crew member on Mercenary ship (Memory Alpha: TNG episode named "Gambit")
- Picard was going by the name of Galen - serving under Baran. This would have been his official position, performing archaeological tests under the oversight of Tallera. The position may have been untitled, but his role and duties were clear.
- He was accepted as a crew member by Baran, who would have granted him this position (while Starfleet simultaneously had marked Picard's position as being deceased)
- Commander/leader/captain of the smuggler ship the mercenary ship that Baran previously led (Memory Alpha: TNG episode named "Gambit")
- After a mutinee that killed Baran, Picard gave orders for where the ship should go next, and they were followed
- Defending Legal Council (Memory Alpha: The Measure of a Man)
- Assigned by Captain Louvois, from Judge Advocate General's office on Starbase 173. This assignment came after Picard referred to some legal regulations
- Basically, officers were used as legal council. As a result of this official assignment, Picard essentially acted like a lawyer/attorney defending Data, arguing for Data's sentience. (Riker didn't like his related official assignment of taking the opposing side.)
- Prosecutor (Memory Alpha: TNG episode named "Devil's Due")
- He went against Ardra, a woman who claimed to be the devil. In there, he challenged Ardra's claim, so arguably he was the prosecution.
- There are other times we see Picard in a courtroom. This one may be particularly noteworthy, compared to other court scenes, because the position he held here was fulfilling a role in the Ventaxian judicial system, rather than being part of Federation Justice.
- The Picard (Memory Alpha: Who Watches The Watchers TNG episode))
- Referred to as "The Overseer" earlier in the episode, Picard's name is revealed to a village who then bestow this title onto the supreme deity.
- I had an inclination to not count this, as this title was bestowed by the town's official timekeeper, a position with no clear official authority to determine new supreme deities.
- However, the timekeeper's actions do seem widely accepted in this small society which is willing to operate on consensus. Lacking a more formalized structure, the fact that multiple villagers perform multiple actions (standing watch, hunting an escapee) indicates Picard's official title did have both recognition and support by the locals. This seems to be how the village seemed to operate.
- Further, "The Picard" became a title that was recognized by the local governing structure, and gave Jean-Luc a unique role if The Picard would choose to communicate with his follows (which he eventually does).
- Best Man (Memory Alpha: Star Trek (TnG) Nemesis movie)
William and Deanna Troi's Wedding
Presumably assigned by the groom (based on current Earth tradition, which may not have been how this was assigned). If so, this title as assigned to Picard by someone who had a superior official position in the wedding.
This led to Picard doing the right thing of giving a speech. So, this official position led to him having a title for the role, and he performed a related duty (speaking) on-screen.
However, consensus seemed to be leaning in the direction of this not counting (presumably because of the short duration). I wasn't so eager to increase the overall count, so I've moved this to the status of not being one of the counted positions.
I propose also accepting:
- Officer, having just graduated from being a cadet (Memory Alpha: Tapestry.)
- Picard refers to being an officer, having recently graduated from cadet status, waiting for first assignment. (Starfleet would have assigned Picard in this status.)
- Despite being seen in a reality created by Q, this was a historical re-creation and Picard was actually stabbed in the chest, so this wasn't just Q's fiction/manipulation, so I think that solidly counts. Unlike other Q manipulations, when Picard first sees this (near the start of the episode), Picard didn't question its authenticity at all.
I've thought of several other roles, that I determined can't be countable:
Ambassador (future, Memory Alpha: All Good Things) (if Worf's Klingon governor of Haatoria and High Council Member from the future counts, so should this. However, my take is that many people consider the future to be a Q manipulation, so based on the original post/question, I'm thinking this should not be counted.)
Commanding Officer (Memory Alpha: All Good Things... has a Q-corrupted version of this seen on-screen)
Just prior to gaining control of the Enterprise, we see Tasha Yar near Picard.
Furthermore, "commanding officer" may be a description, rather than a clear position/rank. Picard's rank may have still just been "Captain" the whole time, so I've determined this does not count as a separate title.
Champion of April 2323 Starfleet Academy Marathon (mentioned (Memory Alpha: The Best of Both Worlds Part 2), but we don't see Picard in that role on-screen)
keynote speaker of 2367 Federation Archaeology Council (I'm not offhand sure if this was on-screen at all, so at least for now I'm not counting it)
Child (Memory Alpha: Rascals) - not really officially assigned position
Captain of the Stargazer (Memory Alpha: "The Battle")
Although this is seen on-screen, it seems that the actual on-screen footage of this position involved a Ferrengi mind-control device. As such, this fits in the "acting captain" category of being rather temporary by nature in that episode. Although Picard clearly had this official position, I'm not offhand aware of this showing on-screen in another episode, so far now I'm deeming it non-counted.)
Memory Alpha: The Inner Light - Anything related to this episode, where Picard lives many years in an alternate life on a planet. Although from Picard's experience, he likely had multiple positions, and they were on-screen, it was a fantasy that conceptually was quite similar to a Q storyline, which the original post/question indicates isn't counted. In our real-time, Picard was unconscious only briefly, so no real official position was granted.
I would think that Picard, noted as an accomplished diplomat, has likely had several other honorific or functional titles in ambassadorial/diplomatic/negotiating roles, but these are just ones come up with so far. (In truth, I'm rather pleased with the variety of official positions sources/authorities from whom some of these official positions were granted.)
Edit-related Notes: This answer is not fully applicable to the current phrasing of the question. Most of this answer was initially written for an earlier version of the question. Later the question was modified in a way that specifically excludes some of parts of this answer, including the two that I felt most happy about. Rather than gut my favorite parts (which showed some creativity during answering) to leave a less amusing hollow shell version of the answer, I'm choosing to largely leave this in tact, even though some people (including the question asker) seem unlikely to agree with interpretations on whether certain posts should count.
Since I believe many people read this site for entertainment, I'm planning to leave this fun answer to continue serving its desired purpose.