My definition is simple, someone that plays for a fee.

Generally, this definition is used to differentiate an 'amateur' from a 'professional' for the purpose of competition, such as in a 'pro-am' sports tournament ... it has little if anything to do with the ability of the individual ...
When I was competing in accordion competitions, if you were being paid as an accordion instructor, or were getting payed to play gigs, you could not compete in the ' amateur' categories ... you had to compete in the 'open' categories ... Believe me, there were 'amateurs' with far more talent and ability than a lot of the 'professionals' ...