I find the biggest stumbling block between "psionics" and "magic" is whether or not (general) you personally use "magic" as a synonym for "supernatural power" or not.
If you do... then "magic" is just the overarching default word for anything in the fantasy world that is "not beholden to physics" (from our real-world perspective of physics, not that anyone in-world and in-game necessarily might know what 'physics' as a concept actually is.) In this case... psionics is "magic" because because it does supernatural action just like all other magical classes and creatures do.
If you don't... and you do indeed see "magic" as the Weave/Aether thingy mentioned above as something specific that people tap into to accomplish supernatural feats, then psionics need not fall into it if you consider psionics internal power (of whatever sort you wish to define it as). But if you do fall in this direction... then you will need to possibly also figure out for yourself whether or not Ki and Sorcery do or don't fall into it either. 4E specifically said that the Monk's Ki and "psionics" were the same thing, but 5E has not stated that as of yet. And as far as Sorcery is concerned... it is also an "internal power" thing, but some would say it's an internal ability to manipulate the Weave/Aether, and not whatever internal power psionics is. Whether or not that differential makes sense to you? That's up to you to decide.
For me personally... I fall into the "magic is a synonym for supernatural" camp. Mainly because I don't like the idea that psionics and only psionics gets their own "thing", and every other class falls into "magic". To me, psionics isn't so important that they get to be the special snowflake, while Arcane, Divine, and Primal "magics" are all stuck under one roof. If Psionics gets to be its own thing, then Arcana, Divina, and Primala should all be their own things too.
(And on a similar note, this is also why I don't believe psionics should have their own mechanical system apart from spellcasting and spell slots... because if they deserve to have their own system, then the Divina and Primala classes should get their own individualized mechanics to separate them from arcane spellcasting as well.)