I have no familiarity with the 2e Psionicist, which seems to be the main inspiration for weird psionic mechanics: something like a skill system?
I'll try to sum it up.
Psionic powers in 2e came in six disciplines: clairsentience, metapsionics, psychokinesis, psychometabolism, psychoportation, and telepathy. Metapsionics was a more advanced thing that mostly had powers that made other powers work differently. Within each discipline, powers were divided into the more powerful Sciences and the less powerful Devotions. Level generally didn't matter much for power use, though some powers had level as an additional prerequisite (this was particularly common with metapsionics). It was however common for powers to have other powers as prerequisites, which acted as something of a roadblock to the more powerful ones, but only for a few levels. Psychokinesis, psychoportation, and telepathy in particular had a large portion of their powers gated behind one particular Science (Telekinesis, Teleportation, and Mindlink, respectively). Psionicists started with access to a single discipline and a small number of powers (IIRC, one science and three devotions), and gained more with higher levels. The discipline chosen at first level would be your primary discipline, and no other discipline could have as many sciences or devotions as that one had. This combined with the gating meant that for the most part you had to be pretty high level to get access to most Sciences in your secondary disciplines. If a non-telepath wanted the Science Domination for example, they would first need another Science Mindlink. And that's two Sciences in telepathy, which means they need at least three Sciences in their primary discipline, which is a total of 5 which can't be had until level 9.
Activating a power would require a roll: d20 equal to or below your Power Score for that power, which was equal to one of your ability scores (most commonly Wisdom but fairly often Intelligence or Constitution, the latter particularly for psychometabolism) plus a modifier (usually negative). Some powers would also have other modifiers: Telekinesis, for example, would be more difficult and more costly the heavier the object you were trying to lift. A success meant you activated the power which then took effect, and you had to pay its activation cost in Psionic Strength Points (which increased mostly linearly with level). A failure would cost half the PSPs. Most powers could then be maintained for a lower PSP cost each round (or occasionally longer) – they almost never had fixed durations, but were around for as long as you kept maintaining them. PSPs would continually regenerate based on your activity level, on the scale of hours. At higher levels, it was quite possible that a night's sleep would not suffice to regenerate your full complement of PSPs (depending on your stats, this would happen some time between level 8 and 10). On the other hand, a similar limit applied to casters who required long times to memorize their spells: a 10th level 2e specialist wizard would require 9 hours of spell memorization (after sleeping) if they had expended all their spells the day before).
Telepathy in particular worked a bit strangely. Most telepathic powers first required you to use the power Contact on the target. Against non-psionic targets (and wild talents), this was a fairly easy affair, just make a roll for a fairly high power score, pay some PSPs (based on target level/HD IIRC), and you've made contact. After that you could use most telepathic powers freely on the target, and many did not grant saves. This was a fairly strong aspect of the 2e psionicist, but somewhat limited by the need to take a round to make contact first.
Psionicists and psionic creatures, on the other hand, had "closed minds", so a regular Contact would not suffice. That's where psionic combat came into the picture. In order to breach a closed mind, an attacker would need to use attack modes: a set of five particular telepathic powers. These had to be selected as normal powers, and in addition to their use in psionic combat they also had effects when used on an already contacted mind. Psionicists also gained a number of defense modes for free that would help them withstand psionic attacks. Using psionic attacks and defenses would of course cost PSPs (except for the Mind Blank defense mode which was free, but pretty weak). In order to breach a closed mind, you needed to establish three "tangents", each of which would require an opposed roll of power scores. The attacker would gain a modifier to their power score based on which attack was used against what defense, in a sort of rock/paper/scissor system. The opposed rolls used what are commonly referred to Pendragon rolls: highest roll that's still a success wins.
There was also a later version of the psionics rules in the revised Dark Sun boxed set as well as Skills & Powers. These were based on the same principles for the most part, but differed in important details (psionic combat being the most important one, with tangents replaced by having to burn through your opponent's PSPs, which was kind of difficult given that most attack modes cost the attacker more PSPs than they would deal "damage" to the defender).
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That's about as much detail as the boards can take, I think. All of that said, I am not particularly fond of the
mechanics of 2e psionics, which to a large degree felt like they had been adapted from some other, more skill-based game. They were wonky, and often highly limited (needing to roll against Con-3 in order to activate your healing power is a bit of a bummer), and with rare exceptions kinda weak, particularly given their linear nature (PSPs scaled linearly with level, and most powers didn't really have stronger effects at higher levels). What I want is the
feel: something distinct from magic, and something that feels cerebral and enlightened. They should feel like a mind that has achieved a higher state of consciousness, not a mind that has been corrupted by foul influences (I'd be willing to entertain the latter as an option, perhaps connected to particular powers, but not as the default mode). Ideally you should also have the feel of a small number of core abilities that manifest in different ways: while Awe, Daydream, ESP, False Sensory Input, and Inflict Pain are all mechanically distinct powers, they
feel like you're using the same telepathic ability to achieve a particular thing.