Psionics and Esotericism

One might distinguish, in imaginative fiction - four positions regarding the existence or otherwise of non-ultratech-generated supernormal powers:

Scepticism/Physicalism. There is no unambiguous and repeatable evidence for powers of any kind. no siddhis, no ch'i, nothing. Whilst there are reports and anecdotes galore, none have ever been conformed. Everything else given a mundane explanation - NDE is nothing but hallucinations of oxygen-starved brain, cases of telepathy can be explained as subversive nanite cybertelepathy, and so on. This is the position of a great many physicalist sects and ideologies, and is widely held in a great many civilizations. Critics charge that this stance selectively ignores or reinterprets a huge amount of factual material.

Supernaturalism This goes in the reverse direction by acknowledging the literal existence of all the beings of Fantasy and Supernatural fabulism and religions - magic, ghosts, goblins, elves, trolls, vampires, werewolves, the lot. This could occur with either:

a hypothetical (and usually quasi- or pseudo-medieval) setting, like Tolkien, David Eddings, the TV shows Xena and Hercules, RPGs like Dungeons and Dragons, etc,
or as a single (or a very few) supernatural element or being (vampires, undead, etc) crossing over into or existing in our otherwise normal world, a la Bram Stoker, Steven King, Dean Koontz, Ann Rice, Peter Hamilton's Night Dawn trilogy, etc,
as a sort of half-half crossover with fantasy and realism given in equal portions - as in Shadowrun, Buffy, Charmed, etc
as Magical Realism which is like a single supernatural element that is added for literary purposes (but could probably just as easily be re-written (with some modifications) as a realistic phenomenon (e.g. Jeff Noon's Vurt)

In any case this assumes the rules of the physical universe are very different, either because this is another universe (Tolkien), or because the physical laws of our universe of the Earth alone has changed (e.g. Shadowrun) or simply suspension of disbelief to tell a good tale. The emphasis is on religion, mythology, or pop-occultism (e.g. Gile's big dusty books [in Buffy]) rather than on science.

Psionics While the laws of the physical universe are believed to be unchanged, and the emphasis is on science, there is still, added to this, a sort of exaggerated psi-element. i.e. there may be a particular race that has extraordinary psi-powers, or, more commonly, a mutation or mutations, so those humans (and other races) who have that mutation have this psi ability. The psi ability is also - to a greater or lesser extent - able to be manipulated by the person possessing it, as easily as one might move one's physical body or practice an exercise in mental visualisation (perhaps a better example). Examples include Babylon 5, Traveller, E.E. Smith's Lensman series, the Marvel Comics universe (the X Men, etc) and so on.

Esotericism While supernormal powers have been reported to exist (but rarely proved under falsifiable conditions), these are generally only attained after long discipline in yoga and other techniques of psycho-spiritual transformation. Often those who have developed these powers ("siddhis") are reluctant to use them, very often the powers are not deliberately sought but area side-effect (even a distraction) of the spiritual quest.

Also included here are

techniques of working with ch'i, as in Tai Chi. martial arts etc, which may or may not be purely a physical phenomenon (again, it is up to the individual to decide)
healings - which may or may not be placebo effect on the part of the person being healed (and not everyone is healed, which adds to the confusion/doubt on this matter)
Out of Body, NDE, etc, experiences, some associated with valid (or seemingly so) clairvoyant knowledge
visions and reports of gods, angels, daimions, etc that are not ultratech powers or archai (sometimes these may be associated with being given healing powers, etc)

In this option, there is some instances of genuine, mostly low level phenomena, a large body of material stuff associated with advanced psycho-spiritual development or unusual states of consciousness, and with more dramatic stuff reported but the interpretation remains controversial or ambiguous (some accepting it, others not), and a good deal of rumour and folklore based on hearsay.

While truly dramatic powers and siddhis are also reported to exist, and even phenomena like psionics, this is more anecdotal - there are some who have - they claim - witnessed or experience it first hand - but it is never enough to convince even a mild sceptic