Well, I'm not knowledgeable enough in Shinto folklore to say that with certainty, but I think oni is an umbrela term, and there are lot's of different kinds of oni.
That being said, I won't hold it against you, as I'm not sure, and neither will the game try to be accurate with that sort of thing. That being said, Kishin is either a more distinguished name and also has the precedent of a character in the series calling himself one; besides, I'd say it's a cooler name than oni.