A professional photographer, in my opinion, is one who does it for a living, or at least makes money from it. It can be a second job, but it has to be a paying job. It does not require formal training, although that's usually associated with it (and for technical things, technical training is usuallly required).
For instance, I'm an electronics technician. I have no formal electronics training other than what I got on the job. I just started studying electronics one day when I was about 15, becuase I wanted to be a ham radio operator. I wound up, without even having high school under my belt, much less a degree, being an electronics technician who works on air traffic control systems. And don't worry, I'm a good electronics technician, so you're safe in the air.... Honor Graduate from my class when the army sent me to the tech school.
Which brings up another point, one I won't get into here, because I will rant about it. I'll just say the point: dropping out of school was the most singularly dumb thing I've ever done.
In any case, I get paid for being an electronics technician, and it's what I do for a living. I'm an amateur radio operator, which as far as electronics goes, is virtually the same as what I do for a living: I repair communications systems when they break and align them when the don't work. I have to, becuase otherwise I can't use them to carry out my ham radio communications. But I don't have to talk on the radio to earn a living, I just like to do it. So I'm an amateur there.
Another side: come this festival here on post this July, I intend to pitch in at the arts & crafts booth my unit is renting, and stack up some prints for sale--assuming I can get any prints worth stacking up for sale. However, even if they sell like wildfire, that will not make me a professional photographer. I earn my bread from electronics, and photography is just a hobby.