The Great Big Lie
Psionics are a lie.
Well, they exist, as far as everyone can tell, but they shouldn't. A psion can't really teleport, they're just very good at lying to the universe about where they are. Psionics are a collective fever dream, a lie just believable enough that we can all go along with it. Sure, a psion can teleport, but only because they can convince us they can.
This is why psionics are magenta. Psions are liars, just like the false color that paints their actions.
Are You Psychic?
When you create your character, for each of INT, WIS, and CHA you have above 15, you have a 10% chance to be psychic (up to a 30% chance). This is rolled once, at the end of character creation, and it is a "yes or no" thing. You are psychic, or you aren't.
If you are psychic, you know the Mind Blast power, and its associated passive trait to understand psychic songspeak, and have 2 PP plus 1 PP for each of INT, WIs, and CHA you have above 15 (minimum 3 PP). You gain 1 additional PP for each psionic power you learn. You regain PP when you sleep, same as spell slots.
Anyone can be psychic. Even a barbarian. Imagine how horrifying that'd be.
Using Psionic Powers
A psionic power consists of an activated effect that spends PP to active, as well as a passive effect that is always active. Using a psionic power takes your turn (or action, or whatever) and is treated as a magical action if necessary (but isn't effected by anti-magic). Consume the amount of PP listed by the power to perform its effect. Once you run out of PP, the universe is done with your bullshit and refuses to let you lie to it anymore.
Learning Psionic Powers
Whenever you witness another creature use a psionic power, you have a 1-in-6 chance to learn it. This effect is cumulative, so the second time you see it you have a 2-in-6, and so on.
Once you learn a psionic power, you must meditate to unlock it by expending 1,000 xp (or system equivalent). Once you do, you gain 1 PP, the power's passive effect, and can use its active effect.
Psionic Powers
Art of Mind Blast
Passive: You know songspeaking, the psychic language shared by all psionic beings. It is an incomprehensible song to all other beings.
Active: Spend a PP to attempt to destroy a creature's mind. Creature saves vs. spells (or intelligence or whatever) or takes 1d6 damage per psychic's level. Only works on intelligent beings. Beings of animal intelligence take half damage.
Art of Longspeaking
Passive: You can communicate telepathically with creatures within 10 feet of you. This speech is two-way, but requires a shared language.
Active: Spend a PP to form a long distance telepathic bond between you and a creature. You can communicate telepathically up to 200 feet away. This speech is two-way, but requires a shared language.
Art of Jaunting
Passive: Your feet lie about their position at all times. You can stand on top of liquids, or thin air, or anything really, but if you attempt to take a step you will sink.
Active: Spend a PP to disappear and reappear in a spot within 30 feet. You don't teleport, you just stop being where you are.
Art of Telekinetic Weapon
Passive: You can touch a weapon to cause it to float by your side. It's covered in a thin magenta mist, and hovers around you, never in the way but always there. Doesn't do much except look cool.
Active: Spend a PP when you fail to attack something to try again with your floating weapon. You use all the traits of the floating weapon when making this attack (enchantments, damage, cool magical effects, etc.).
Art of Push-Pull
Passive: You can grab objects up to a foot outside of your actual hand's reach. You still have to hold them, you can just do it further away.
Active: Spend a PP to attempt to move a creature of object. If it's an object, it's moved up to 15 ft. in a direction of your choosing at extreme speeds. If a creature, it must save vs. paralysis (or dexterity) and if it fails it's shoved 15 ft. in a direction of your choosing at slightly less extreme speeds.
Art of Other-Self
Passive: You can make small modifications to your appearance at will. You are still obviously you, but you can make dirt disappear and remove unseemly blemishes. Or add dirt and blemishes if you so desire.
Active: Spend a PP to create a duplicate clone of yourself. This clone acts like you and performs the actions you would in any given scenario, but is incapable of making meaningful impact on the world. It's attacks do no damage, it's hands pass through objects, and everything about it just seems a bit off. When everyone stops believing in it, it disappears in a cloud of magenta smoke.
Art of Easy-Truths
Passive: You have preternatural control over your voice. You always sound like you wish to sound, and the meaning of your words is easy to follow. You can also change your voice in cool ways, like making it louder or echo.
Active: Spend a PP and with a quick wave of your hand while speaking a lie, you convince a creature that your lie is 100% truthful. They'll figure it out if they are presented evidence that obviously disputes your lie, but otherwise they accept it as truth.