This feels appropriate

Except I’m not talking about what he’s called, I’m talking about his character.

While that’s true, it took a few writers before he really came into his own. It was the 2000s before he was the meta, witty, merc with a mouth. The parody was a lot more on the nose and it traded some of the parody for the meta, witty Wheaton-isms and pop culture references. Parody Deadpool and Deadpool Deadpool are arguably different characters.