It depends on the depth of the chair: if soft and deep, so the sitter sinks into it, then the proper preposition is in ; if flat and hard, so the sitter perches on top, then the proper preposition is on.
"Seat" is a noun that refers to a place or piece of furniture designed for sitting, while "sit" is a verb that describes the action of lowering oneself into a sitting position. For example, you could say, "Please take a seat," using "seat" as a noun, or "Please sit down," using "sit" as a verb.
