I'm on the board for a co-op (housing co-op, not worker co-op) that's incorporated as a non-profit. I know we've talked about how we could have incorporated as a cooperative instead, and I think we could have been a cooperative non-profit....we just weren't.

Co-op is about ownership (and the seven principles, but legally I think chiefly about ownership), whereas I don't think non-profit status is, for the most part.