Jirachi being able to defend itself means that it has strong mental powers, another attribute of being psychic.
Celebi can actually time travel instead of just leaving this plain of existance, going to the fairy realm, and then coming back (See Spiky-eared Pichu and it's time as, for lack of a better comparison, Celebi's companion.). So even if it is based on a wood nymph or a fairy...it still gives off the psychic feel because it can still open up wormholes whenever it wants and move forward and backwards in time.
Landorus, they seem to favor their kami counterparts more so then genies.
Not to mention Victini really doesn't favor fae at all besides being overly cute and created to appeal to girls like the stereotypical fairy.