I think, in general, you are considered a professional photographer if your primary source of income comes from your photographic work. For example, if you are a wedding photographer by trade, your job is to photograph weddings. You are a "professional" wedding photographer. The same would be true if you were a sports photographer, and sold your work to various sports-related franchises, newspapers, magazines, etc.
On the flip side, you would generally be considered an amateur photographer if you simply do photography as a hobby, without making any money on the deal. Other common terms for this are "hobbyist" or "enthusiast". It is possible to be considered "semi-professional", in that a portion of your income is earned from photography, while your primary profession is something else.
Fundamentally, I don't think formal education really has anything to do with being a professional or not. I think that many professionals are formally educated, but I know some photographers who do sports or wedding photography professionally, and they simply picked up a camera one day and started learning. They have no formal education, but have some phenomenal raw talent.
I think it is important to note that, like you mentioned yourself, the quality of a photograph has nothing to do with whether you are a "pro" or not. The quality of a shot ultimately boils down to the individual taking the shot, their skill/talent, their work ethic, their diligence, and their sense of artistic vision. None of those things require any kind of formal education, nor do they require that your photographic work be the primary source of your income.
That said, if you make your living via photography, you will undoubtedly become more skilled than someone who casually does photography, simply because of the sheer volume of shots you'll likely take, and the time you'll invest in using your camera and its settings, in post-processing, in working with prints, etc.