In terms of 5e casting, in your mind, how exactly are Wizards different from Clerics or Bards?
They all have Foci or Component pouches. They all have the same exact spell slots. Bards are different in that they don't have to prepare spells... and that's it mechanically.
Spell lists are different, sure. But other than that they are pretty much exactly the same.
Nah, class abilities impact their spell casting. There's a big difference between having arcane recovery, spell mastery, signature spells, and subclasses abilities that enhance spell casting versus not having those things, on top of different spell lists and ritual mechanics. Spell casting itself is a similar mechanic but that's about it. There are plenty of other differences.
Also I noted to my surprise these Psionic spells were on the Bards spell list too, which suggests it's College of Eloquence might be Psionic in nature as well, calling back to the 3.5 Ardent class.
They are mostly enchantments and abjurations. That's the type of spell one would expect on the bard list regardless.
2. If psionics Are going to be considered as another arcane "school", does that mean that other arcane users can grab a psionic spell or two? I can see that being a lot of fun, but it definitely eliminates any particular "specialness" of psionic characters. Especially if there's not going to be a reworked mystic/psion/whatever.
Those are spells on the standard arcane spell lists. Eldritch knights add abjurations and evocations so mental barrier, thought shield, and psionic blast are readily added, plus more based on the non-restricted levels for adding spells. Arcane tricksters add illusions and enchantments so id insinuation, mind thrust, and ego whip are also readily available; and there are levels where other schools can be selected.
I think that was the point in not making the psychic warrior or soul knife as spell casting subclasses. Those existing subclasses can already cover the psionic caster subclass and flavored as such by the players.
It's not the spells that make psionics unique. That overlap usually covers a lot of spells shared with psionics and the UA article included several spells that already fit the flavor. Psionics as another magic specialty takes away from the possible unique nature of the spells / powers but the subclass abilities add that in and are not normally available to other subclasses.
This creates the odd situation where the Divine Soul Sorcerer can wield Arcane, Divine, and Psionic Magic.
That's not really a thing. Spells themselves are not divine or arcane and have not been for some time (before 3.0). How a caster manipulates magic determines if the caster is arcane or divine. Artificers, bards, sorcerers, warlocks, wizards, arcane tricksters, and eldritch knights are arcane casters. Clerics, druids, rangers, and paladins are divine casters. That doesn't change the nature of how they manipulation magic even if the spell exists on a list from both categories of casters. EG cure light wounds being cast by an arcane caster is still just cure light wound; blindness / deafness cast by a divine caster is still just blindness / deafness.
XGtE states the divine soul has magic that is linked to a divine source. That would mean all spells the divine soul takes, not just the ones on the cleric list.
Spells being classified as either was changed a long time ago at this point. Designating individual spells as arcane or divine is a house-rule label at this point. Not that people don't do it because it was definitely a thing in the past and typical of many novels as well.
Just my 2cp.
And also the way they did the spells Soulknife and Pyschic Warrior are not potentially the only Psionic subclasses for the Rogue and Fighter, because both can cherry pick Psionic Spells via Arcane Trickster and Eldritch Knight.
It's not really cherry-picking. Magic isn't any more proprietary than skill proficiencies are at this point. The types of spells typical of psionics just coincides with the types of spells these two subclasses already normally take. That just stems from the psionic-equals-magic approach the UA is using instead of the psionics-does-not-equal-magic approach we've seen in the past.
Psionic spells are not special, and that might not go over well with a lot of people. The special part comes from the class abilities so hopefully the final product works well in that regard.