Technically a professional musician is one for whom playing music is a profession, in essence, how they earn their living. This could be all or part of their living.
It's difficult for many in this day and age to make a decent living without some side hustle.
If one is a bank clerk by day and waits tables at night, which is their profession? Actually both, and it cannot necessarily be the job that pays the most because the waiting tables could pay more at times that others.
One may say this is my profession but I do that to make ends meet.
If we start tossing "semi-professional" around, that's just splitting hairs.
The word professional has come to mean other things in music and other fields. It has taken on the connotation of the attitude of how one conducts onself, in a professional manner, whether a dime is being made or not.
It's like the word "gentleman" which once stood for a certain class of person, not really a fellow who holds a door open for a lady or uses a specific restroom.
Such as it is with language, constantly evolving, thus you see ads seeking musicians, must be professional... you check into it and find it doesn't pay, but they want the dedication of a professional. Unfortunately for most who seek this person is that mainly the motivation that makes a professional have such dedication is the money.
If one plays for any money at all, that is a professional, at least for the gig that paid.
If one plays for sheer enjoyment, that is an amateur, which by the way doesn't equate to "novice" anymore that professional means "expert".
I think people like umbrella words that say something about a person. A person shows up on time, learns the material, works well within a band, does her/his homework, doesn’t get drunk and naked on stage; all this has the markings of what a professional would be like, so we say that’s a real pro, rather that saying they do all that other stuff.
There are many well known musicians who do not conduct themselves in what has come to be known as a professional manner, and they earn a fortune. They stay drunk most of the time, do drugs, get caught up in scandalous activities, yet they pack out concerts and sell cds.
It’s how you look at it... if one thinks they are a professional because they step up and act like what they believe it takes to make a pro, let them claim that.. walks and quacks like a duck...