What is the difference between magic and psi? Some games, like D&D, seems to consider them differently

Trying to describe the differences is tricky, what with the 'rules' behind them differing depending on setting.

If magic is reality's cheat codes, psionics is being able to alter reality's programming. One works with the existing framework to make things happen, the other is you going 'fuck you I do what I want' until the universe gives up and lets you do what you want.

So psionic is more like turning all the iron around you in a sword to stab things in the face, while magic is spawning a new sword out of nowhere?

Mostly everyone else in the thread is correct, but I'd like to throw in chi and divine powers as another couple concurrent systems.

Generally, the two factors of supernatural systems are availability/rarity and source of powers. Rarity refers to how often random people have the ability or can express it. For example, usually everyone is capable of learning magic or training to use chi, but generally psionics are born with power and divine powers are gifted directly by higher beings. This also brings up the second factor- the source of said power. Both chi and psionics stem inwardly while magic and divine powers must be obtained from an outside source. You could even set up a little chart to mark the differences:

Magic: Learned and External
Chi: Learned and Internal
Divine: Given/Born and External
Psionics: Given/Born and Internal

D&D only considers it differently because they were trying to sell more splats back in the day.
Nowadays the only difference is narrative due to everything having to go through spell resistance for streamlining, and psi shit being based on point pools rather than spell slots.

What's the difference between using a robot to pick up a cup and picking up the cup?

And if you'll pardon the anime references, both Boku no Hero Academia (MHA) and One Piece (OP) each use a different system with their powers, Quirks and Devil Fruits respectively.

Quirks in MHA are abilities that people are born with. They can vary quite widely from person to person, but generally are described as mentally performing similar to a muscle- they have to be trained and overuse/overextension can damage them. Normally these rules would squarely place Quirks in as Psionic abilities, if not for the fact that over 80% of the populace has a Quirk, skewing them much closer to being similar to Chi in avalibility.

Devil Fruit powers are either elemental, beastial, or thematic, but they all stem from eating it's corresponding Devil Fruit. There can only be one of any kind in the world, and when a user dies their power is recycled back into a random fruit. Additionally, the range of powers a fruit will give you is extremely limited, meaning that you can't really branch out of the one trick pony your fruit turns you into. Normally this would set Devil Fruits as clearly Divine (or, well, demonic) powers seeing as they are external given powers, but a Devil Fruit can just be found and eaten by any person who finds (or buys) it, and many fruits have a deceptively high skill cap, encouraging a user to think creatively on how to use their power. For example, Luffy with his Rubber powers can essentially naturally blood dope, since his arteries and veins are rubber and thus can pump much more blood much faster than a normal person would survive.

So there are a lot of ways to play with power systems to effectively function as all of them, as long as you're not already allowing multiple ones to begin with.

Let’s not forget that psionics have a thematic connection to chi; if we are to look at the kind of training and awakening that someone like The Shadow went through to gain his abilities. Associating psionic power also with meditation and eastern philosophy alongside concepts of chi, ki, or qi.

Then there are thematic instances of one having their potential for it blown open by exposure to things like radiation or the unfathomable existence of beings beyond our ability to fathom.

The irony in this is that magic is more scientific than the fucking science-fiction trope of psionics.

Magic: You explode violently on whatever has the misfortune to be in your way, but then you must take a while to recover.
Psychic Powers: Your results are fairly nasty, but they kinda plateau early. BUT, you can fairly reliably spam them with minimal downtime(unless you fail to activate the power, then you might need a day or so to rest).
>Summoning: You need to take your time, set up the circle, light the candles, scribe the runes, and burn the energy you'd normally use for magic, but the results can damn sure be worth it, either in the form of a new Pokemon in your collection, you taking on the form of an ancient hero, or cheesing people with Final Fantasy Tactics-style summons.
Ki: Make your own Spirit Bomb/Kamehameha? Check. Shadow Clones? Check. unleashing the Third Impact by flexing kinda hard? Check.
Nemesis: See those guys above you? Fuck. Them. They're never getting to do their fun stuff, because you fucking said so. Also, channeling undead-style energy makes you really tough... Who'da thunk it?

>chi, ki, or qi
It's just "chi". All the spellings of the word produce the same sound: 'kee'.

Thematicay, what we usually consider "pop fantasy magic" usually includes creating stuffs out of thin air or inflict change: throwing fireballs, wielding elemental powers, external summoning, curse, shapeshifting, divination and potion brewing.

Psychic powers are usually more about abilities that let you perceive/experience things that wouldn't be normally possible: Telepathy, mediumship, remote viewing, mental projection, pre-and-postcognition, psychometry etc.

There was a period of time where scientific inquiry was hand in hand with mysticism. Studies were done to try to prove or develop psychic ability in individuals, namely precognition. It was a hot topic of fringe science which became a seedbed of inspiration for science fiction authors. It's since been grandfathered into the sci-fi genre but expanded in scope from the possible capabilities of the mind, to just straight up magic by another name.

Typical Magic in games and fantasy tend to have a set of rules for them of what they can and can't do. They often have spells, which have requirements to perform, like incantations and ingredients. Psionics don't have these kinds of trappings to them, since the power comes from the mind.

However in the terms of a game, they need to have those sorts of trappings, like named abilities and costs, otherwise they can be exploited and imbalance the game.

More like Magic: You harness an external magic through learned motions/combining ingredients/channeling power from an vessel (wand, etc) to turn a piece of wood into iron. While psi would be willing a piece of wood into iron through pure willpower with your special snowflake brain mutation that takes zero training.

OTOH, Chi at is "early" form (as in Wuxia novel stuffs, without the influence of Dragon Ball/Elemental Bending), is all about bodily function regulation/enhancement, such as punching/kicking/biting/headbutting really hard, longevity, ability to not breathing for extended period of time, temporary skeletal deformation to cram into really small places, high jump, makes your body lighter, superb speed, coordination, precision and balance, as well as temporary shut down certain organ/stop blood flow on specific part of the body/contract muscle on specific part.

Magic hurts when you do it wrong, psi hurts when you do it right.

Psi stuff originated with more mystic magic shit but over time got co-opted as NOT-magic for everyone that wants to pretend they're aiming for a more scientific theme.
Hellboy has some of the older pulpy themes mixed with some more modern conceptions.
I'm sure you could find examples to the contrary like when it's more treated as a straight-up superhero-style superpower but psi stuff is often associated with more creepy paranormal shit like your pic or other stuff with ghosts and the spirit world which is just another of the many places where the lines are blurred with magic even including demons in some cases.

At the end of the day it's magic but more specific and themed magic with a focus on stuff like extrasensory abilities. D&D just has another mishmash of pulpy and sci-fi old and new trying to be different just to be different and getting buried under "normal" d&d magic's better and more numerous options. The 3.5 psionics rules were basically just recycled spellpoint alternative rules.

Psi is spooky.

Psi is essentially where you gain so much direct mental control that you can mentally control the energy of material around you as if it was an extension of your body.

Arcane Magic is using an external force that already flows through everything and maniuplating it more manually, through words of power, somatic gestures, infused resources, ect.

Divine magic is asking for an all powerful being to manipulate reality for you in some way, either by knitting flesh and healing wounds, or protecting you from the forces of evil, among man

At the end of the day, its all magic, its better if its not explained in depth, but only maintains a logical consistency.

Well, there are also plenty of sources where magic relies on stamina or some sort of internal reservoir of mana or magical power or literally drawing magical power to use from an external source like some magical being or magical force like nature or the speedforce or whatever fits your needs.
You don't really see "mana" associated with psi stuff as much as you do with magic, I can't think of any instances where it is.

Ultimately the distinctions are arbitrary and you're using a certain categorization to create a balanced symmetrical system. That's totally fine if it works for you for your game or world or whatever but it's not necessary or the default.
Chi, divine powers and psionics are all defined as magic in various settings with sources and rules to fit the story and theme. There are general groupings of themes, tropes, memes, traditions, and stereotypes but plenty of wiggle room and variety.

Strawmen aside plenty of sources have psychic abilities take tons of training to use and control.
On the topic of d&d as mentioned in the op it takes just as much training and effort as magic or any other progression.
Mob psycho 100 has examples of both tropes taken to extremes with the main character having immense natural power with the sub-theme of having difficulty controlling it and mostly suppressing it while having other characters spend years of effort and pain and experiments for barely any power.