"you are a musician when your performance takes up
more time than your practice."
Hmmm... I've known some folk who NEVER practice, then go out in public to render (rend?) tunes or songs with instrument and/or voice. Given the above criterion, they are musicians. Listening to them, though, I think the results are different from what openmike implies.
"an amature practices until they can play it without making any mistakes.
A professional practices so much that they will not ever make any mistakes."
I've heard a variant on that... an "good" performer (whether paid or not) is one who can cover mistakes in a way the audience never notices. "There are no mistakes, there is only Improvisation". We never achieve Perfection (at least, not while we are living humans) so the trick is to keep the audience from paying attention to the imperfections.
Personally, I think a Musician is anyone who chooses to create music so others can hear it. "Creating" in this case means making the actual sounds & arrangement of notes, i.e. performing with voice or instrument, as opposed to playing back a recording that someone else has made. Of course, not all Musicians make music that *I* think is "good", but that's just a question of personal aesthetics.