So, what is the basic definition for "trans-warp"? Is it simply any technology allowing a ship to travel faster than the known limits of conventional warp drive? Was Excelsior simply a "next-generation" warp drive ship that pushed the envelope that much further?
Trans-Warp could simply mean 'Beyond-warp' - as far as literal interpretation goes.
And indeed, throughout canon, we do see the term used for speeds that go well beyond regular Warp drive.
From a canonical point of view, it might be implied from the movie that the Excelsior was supposed to have a 'next generation propulsion' known as Transwarp at the time that would allow speeds that go well beyond standard Warp engines - at least, that was the intent.
As we know, Scotty sabotaged the Excelsior and that was the last we heard of Trans-Warp in the 23rd century (as far as my memory goes).
I find it very odd that no canonical explanation was provided.
But we do know that Roddenberry wanted TNG and the Enterprise-D to explore another galaxy as opposed to our own.
This DOES fit well into the premise of a technologically advanced society that doesn't have money and uses full scale cooperation coupled with automated research to advance - but as we saw, this didn't come to pass, because the writers didn't want to make it look like the Milky Way was 'too small' (which I don't really find as making sense to be honest).
In TOS for example Warp speed of 8.4 was described as allowing to traverse about 990 Ly's per 1 solar day - which isn't far fetched by the 23rd century when you take into account what I said above about cooperation, automation and no money.
By the 24th century (or when Enterprise-D launched, I could see Warp drive as increasing in speed by a good amount... for instance, being 1000x faster than TOS Warp drive - allowing the D to get to Andromeda in just under 3 days and start exploring it...
Mind you, the Milky Way would still be under heavy exploration by Starfleet in the 24th century, considering you'd need large amount of exploratory ships to cover it (depending on sensor range, etc.)
At any rate, Warp drive as seen on DS9 and Voyager was way too slow.
I think Paris's statement of 9.9 being 21473 times LS might be accurate, considering that no SF ship on-screen was able to use that ship as a cruising speed.
Voyager technically could, seeing how its maximum sustainable cruise velocity was stated to be 9.975 (which would technically allow 440 Ly's per hour travel - if every increment past 9.9 doubles the speed).
I find it sad that the Federation was not given by the writers Transwarp in late 23rd century.
It would be far more consistent with their progress and technology - but unfortunately, network executives had other things in mind.
Getting back on track...
Transwarp as we have seen has been used thus far by the Voth and Borg.
The Voth used a Transwarp version that was closer (in special effects) to what we saw in Threshold episode of Voyager, but evidently, it was not 'infinite velocity' so much as it was 'much faster than Warp' and managed to cross 60 Ly's in the span of seconds.
The Borg used their own Transwarp - by opening various conduits.
As we've seen, the Enterprise-D was able to open those conduits by figuring out their aperture frequencies (simply because they were already there).
I think that what the Borg might have done here was create 'temporary' conduits... ones that would last specific amount of time before deteriorating unless they are used (though how long would they last is unknown).
The Borg ships later on needed Transwarp coils to achieve TW speeds, and Federation ships couldn't use those conduits without them it would appear (a security feature perhaps?).
As for the TW Hub... that was something which was introduced as a concept in the last episode of Voyager from the writers, however, the conversation implies the Borg had 6 of those in the Galaxy, meaning that the Borg would likely have been using these Hubs as means of transport, though, this doesn't mesh with 'Dark Frontier' where the Borg ships were apparently able to use TW wherever they wanted independently... and indeed, this is confirmed by 7 of 9 in episode 'Inside Man' when describing how a Borg Cube enters a TW conduit.
Unless the Hubs are used as means of facilitating faster travel compared to what individual ships can accomplish... which in a way makes sense if you have a larger force of Borg ships as the Hub that waits to be deployed to various regions of the Galaxy for studying various species for assimilation potential so the individual ships do not expend their own resources (instead, just use the centre of a nebula as a giant power source for the Hub to initially launch the ships, after which the ships only need to expend power on subsequent exploration and eventual return if recalled).
Slisptream is another technology allowing faster than Warp speeds... but it doesn't seem to be classified as Transwarp.
Slipstream was described as being similar to Borg Transwarp... but it also seemed to be better than Borg TW.
V1 of Slipstream as seen in the fake Dauntless allowed speeds of 300 Ly's per hour.
V2 as improved upon by Voyager crew with Borg tech and Benemyte crystals was able to achieve 10 000 Ly's per minute (if you look at the dialogue, the phase variance issues don't start until 17 seconds into the flight, and roughly 1 minute into the ordeal and when Harry transmitted the shut down phase corrections, Voyager managed to traverse 10 000 Ly's at the end of the episode - this didn't take more than 1 minute perhaps in-universe).
Transwarp as a term could have wide-reaching explanations... and realistically, Slipstream falls into the category of transwarp... but I think reclassifications are needed since these propulsion technologies achieve faster than warp speeds using different regions of Subspace (probably).