I've seen this very same debate on the photography forums, and to me, there really isn't going to be any single consensus as everyone has their own opinion on that matter.
The one argument that I always hear will be that if you earn a living from it, you're a professional. Otherwise, you're an amateur.
But then there's another argument I hear a lot is where the pro is very good (i.e. extremely skilled), and that sometimes runs afoul with the idea of a bad musician (or photographer, or whatever) but still earns a full time living on it.
I think one problem is that the terms "professional" and "amateur" can have different meanings depending on context. On one hand, it can represent skill level, but on the other hand, it can also refer to a job (e.g. a working pro vs amateur hobbyist). And the differing opinions happen when people mix or muddy the terms.
Yes, but define "earn a living".
The guy who does NO other work than performing music, and earns all of $25,000 a year - and he's married to an RN who earns $90,000 a year. Is he a "professional"? If they get divorced and he has to start driving Uber and working behind the counter at the auto parts store, but he's playing exactly all the same gigs - does the fact that now he's got to take other work in order to survive, mean he's lost the "professional" status?
Another common scenario for young musicians is the First National Bank of Mom and Dad. If the young musician is doing NO other work than performing music, but his parents bought a condo and he's living there rent free; and they're also paying his car insurance and he's on their medical insurance. Is that "professional"?
Let's consider two young musicians, who pack up and go to New York City to try to make it on the scene there. The first one has no financial assistance from family. You can't survive in NYC as a 22 year old musician just getting started. He's got to take all kinds of non musical work, just to be able to eat. The second one gets an allowance from wealthy parents. He doesn't have to take non musical work. He can focus purely on making the scene, getting known, learning and getting better. Now when these two youngsters are both 25, three years into living in NYC, the first one is still taking all kinds of non musical work, the second one isn't; they're both equivalent talent and skill, and they're both playing the same number and kind of gigs. So the guy that's been getting a check from Mom and Dad every month for three years, is a "professional" and the other one isn't?
The idea or ideal, of the "professional" musican that does nothing but music, is pretty much obsolete. There are a FEW people who can make it without a spouse with steady employment, or support from mom and dad, or teaching lessons, or working at something else. But every year this becomes less realistic.