So. For 5e. What should the lore for psionics be? Is there a cosmology that we can all either love or tolerate?
I suspect we can all agree, psionics is one of the main power sources: martial, arcane, divine, psionic, and primal.
Actually. Arcane and divine are moreorless the same thing, two ways to engage the "Weave"...
I doubt you can get a consistently approved and desired singular approach. I don't think a single approach would get universal support from even half of the people on these boards.
As for what the lore should be, despite a majority of people disagreeing with it, I'd say the best they can do is set themselves up to win people over. The best way to do that is to appease a dedicated group of players that will be willing to pay for psionic related material. One group known to spend a lot of money on books consistently are long term gamers, and the best way to appear long term gamers is to appeal to their sense of tradition and history. To that end, I'd build psionics ... how I built psionics in my setting: Based upon the lore and traditions of old.
Features I'd advocate for seeing:
1.) It comes from internally generated power not associated with the arcane, divine, nature/primal or any other magic based power source.
2.) It has distinctly different mechanics from spells/magic, meaning that detect magic would not detect psionics, etc... and we would not use existing spell mechanics to describe psionic powers. Instead, psionics would be as different from a wizard as a fighter is different from a wizard. After all, both the Battlemaster Fighter and the Wizard have mechanics that can frighten, prone, and push enemies - but they're very distinct from each other.
3.) Uses the 5 historical psionic attacks and 5 historical psionic defenses in ways that are meaningful and useful outside of psionic on psionic combat. This is not that hard, but the crafting would need to be careful to implement without violating rule 2 above and making it seem too similar to casting spells/cantrips.
4.) Allows for 'wild' psionics, potentially via feat or background.
5.) Allows for psionic classes such as the psion and psychic warrior.
6.) Provides classes and subclasses that do not feel like wizards, monks, sorcerers, etc..., but instead have a unique feel that is distinct to the psionic classes, but not anathema to fantasy 'purists' that see all D&D through the lens of Tolkien. You could have 'overlap' subclasses (like the eldritch knight overlaps wizard onto a fighter), but you'd need to have the main class be dominant and the subclass be supportive/flavor. I do think the existing psuedo psionic subclasses approach this idea, but I also believe they can be done better. This can be done - I've won over players that didn't like the idea of psionics in D&D once they've seen it in play in my campaigns (mostly in the 2E, 3E and 4E eras, although 4E examples are less relevant).
7.) Respects and (where possible) utilizes the teminology of old such as the Sciences (Clairsentience, Psychokinetic, Psychoportative, etc...), Devotions, Disciplines, Powers, etc...
8.) Provides alternate versions of traditionally psionic monsters within the 5E rules that utilize psionic mechanics rather than spellcasting mechanics.
9.) Is mechanically approachable to all - meaning it isn't overly complex.