Psychic type is indeed mind manipulation, lad.
Psychic type is good against fighting because mind over matter. Poison types due to some myth about mentally purging diseases from the body. Weak to ghost, Dark, and bug because they're all normal fears.
The Pokémon you say have mind manipulation without being psychic type (Gengar, Ninetales, Lucario, etc.) can all learn psychic moves.
The Pokémon associated with mind manipulation, Hypno, Malamar, and Mewtwo, are all Psychic type.
Confusion and Psybeam are both mind manipulation due to causing confusion.
Amnesia and Calm Mind are things. So is Shattered Psyche.
Telepathy is a thing. Telepathy is still a variation of mind manipulation even though it has its own seperate page.
The smartest Pokémon, aka the Pokémon with the greatest minds (Alakazam, Metagross, and again, Mewtwo), are all Psychic- type.
(Probably my weakest argument) Psyduck's psychic powers give it headaches. What lies in the head?
The conceptual embodiments of certain mental facets (emotion, knowledge, and willpower, are psychic type, further compounded on the Pokémon capable of reading emotions, Gardevoir, being primarily psychic type.