I seem to remember reading somewhere that Anne McCaffrey (or was it Le Guin? it's been a long time since I heard the story) was told point-blank by a prospective editor: "If it has deck plates, it's sci-fi; if it has trees and dragons, it's fantasy." I think that most casual fans have pretty much that attitude about it.
Being a bit of a genre nerd, my own definitions are basically:
Fantasy-As-Broad-Term: Anything where some or all of the rules of reality go deliberately out the window. Thus, "The Twilight Zone" is fantasy, so is Tolkien, so is
Star Wars.
Weird Tales: Stories, generally in a modern or not-too-distant-past setting, where someone in a more or less "normal" world is confronted with the supernatural or at least the extremely bizarre, including ghosts, multitentacled horrors from beyond space and time, telepathic conspiracies, or gray alien proctologists.
Sword and Sorcery: A subset of fantasy concentrating on ambiguous or even dark, relatively low-power heroes, in settings where magic tends to be rare and generally sinister.
High Fantasy/Big Fat Fantasy: A subset of fantasy tending towards long, epic plots with lots of subplots and supporting characters, a lot of magic all over the place, for both good and ill, and usually good or at least well-meaning heroes.
Hard SF: Made-up stories based on accepted scientific theories and/or the plausible extractions thereof. Can have one or two handwavey sorts of things (FTL travel being a common one), but such things are avoided when possible, and the ramifications of their existence are well thought-out.
Space Fantasy: Basically, high fantasy with spaceships and cat-headed aliens. My personal favorite, actually, but pooh-poohed by most editors.
Space Opera: Any kind of massively epic tale set in space. Can just as easily be hard SF as space fantasy.
...
By those definitions, psionics could be an element of hard SF if they were one of the "ground rule changes" and had a convincingly scientific underpinning*. However, they are more likely to be an element of weird tales or space fantasy.
-The Gneech
*ESP and the like has been scientifically studied; the problem is/was/probably always will be that it's not reproducable. You can't prove a negative, so therefore you can't really prove that they aren't there ... but there isn't enough solid evidence to solidly prove that they ARE there, either, at which point science throws up its hands and says, "Whatever, I've got some atoms to smash." Suffice to say, the people at the Cayce Institute certainly believe that it's real and work from the assumption that it is, so a hard SF story that included it would basically start from the premise that they're right and project from there.