Something that captures the flavor of the psion, which has a few key points:
OK, sounds fine. I can work with that provide it's an internally consistent request. Let's see.
If you really think you'll just convince all the players who aren't satisfied with wizards-as-psions with your brand-new, never-before-suggested "just reflavor it!" argument you really have not been reading the room - for 45 years.
Wait a minute? I thought you just needed something to capture the flavor of the psionic. Now you are saying that to capture the flavor you need ... mechanical differences? What does that have to do with flavor?
That's an absolute non-starter. It is and always has been spellcasting. Sure, maybe it's spellcasting by a different means based on your inherent magical ability rather than learning and study, but it's always been spellcasting. Sure, maybe it wasn't originally modelled with spell slots but mana points because originally it represented non-class based magic that wasn't tied to your level (and didn't increase) and so spell slots by level were awkward , but it's always been spellcasting. "Psionics" (how do I hate that word) are just psychic powers, which you will later admit yourself. So if you are going to admit that its just psychic powers, then it's spellcasting no ifs ands or buts.
Again, if really this just comes down to "BUT I WANT MY SPELL POINTS!" or "I HAVE TO BE DIFFERENT!" then I'm really not interested and there are tons of rational reasons for that. For example, if you come to me insisting that you have to have two magic systems that are incompatible and one of them isn't really magic even though its really magic then I have to take account of that when stocking or generating magical treasure. I now have to have magical treasure that just works for arcane casters and magical treasure that just works for psionic casters. And that totally trumps, "I HAVE TO BE DIFFERENT!" because your difference is making the game suck. Likewise, I don't need complicated interactions to debate like does this bypass spell resistance or does it detect as magic when someone uses a psionic power in the presence of a magical sensor. It's all going to be magic, so that it "IS NOT SPELLCASTING" is just not even remotely something to negotiate when designing hypothetical edition.
but just calling a sorcerer a psion while making them wave a wand and chant an incantation ain't gonna cut it
Now that I can be on board with, but really it's just a statement of your second note.
2. Is based on pure mental discipline, not components or special words or items needed.
OK, so first arcane magic is already based on mental discipline so all this really says is you want your "psion" to be able to eschew spell components. You want no verbal, somatic, or material components in order to work magic. And OK, that's a thing, and with material components it's not even a big thing because hardly anyone tracks them anyway. They are mostly there to impose costs on rare ritual magic or to allow wizards to be imprisoned and the like. But, it's worth noting that verbal and somatic components are regular and frequent restrictions on the use of magic that matters in very meaningful ways. Like if you want to cast a spell at a dinner party it's a huge advantage if you can do it still and silent. And what your really asking for is to get away with things if they fail their saving throw because who did that, right? So what you are really asking for here is to be a Sorcerer but better. Not just a little better but A LOT BETTER.
Worse, often what you are really asking for here is advantage on surprise or Quickened magic. With no warning about a spell being cast and no actions to take place to do the magic, what the player is often asking for is the power to always win surprise, to go first, and to take no time doing it. It's like the player that thinks he deserves surprise by describing how sudden his attack is and by announcing it first he won it. And I think that's a massive advantage if it can be argued for and generally dysfunctional otherwise.
I'm not unsympathetic to the request to do most of your magic Still and Silent, but this is a very valuable thing you are asking for and so you are going to have to give up a bunch or else except that things can only be Still and Silent in a very narrow domain in order to prevent abuse. Are you with me on that or is what you are really asking for is just I HAVE TO BE BETTER THAN THE WIZARD! Eh?
Focuses on the kinds of powers psychic are known for in fiction: telepathy, telekinesis, etc.
Yes, but psychic powers are just spellcasting and traditional wizard powers. And the more broadly you define psychic powers like psychic surgery, pyrokinesis, detect spirits, communicate with the dead and so forth the more it's obvious that what you are really asking for is be the wizard but better since you are clearly doing it all anyway but with far less restrictions. So really, there is no focus here or narrowing down of powers. Psychic powers straddle the arcane/divine line because they are ultimately rooted in traditional Western shamanistic and occult practice, just rebranded late in the 19th century when superstition became socially problematic. (See also Theurgy and White Magic in the middle ages for similar but different reasons.)
Ties into the existing psionic lore of DnD, like Mind Flayers and Gith and aboleths.
Sure, that's not hard. In my game you might be like, "Grandfather was once an aboleth slave before he escaped, but he escaped changed and now my family is blessed or cursed with strange powers in their blood and some members of the family manifest strange mental powers!" Totally acceptable backstory.