We so often come back to this question here on AZB, and it's no easier to answer than before. For starters, just like in golf, we have playing pros and teaching pros, and I believe that those who make their primary living from teaching pool have every right to call themselves pool pros.
... but I suspect the question in the original post is about what constitutes a playing pro. Certainly, anybody who derives their primary living from competing at pool is a pro. In America, there may be only about 10 of these. I tend to define a pro more liberally than that, including players who play pro speed. I've waffled on this over the years, but I think pro speed means at least Fargo 725 and, at times, I've even put that number as high as 740.
These days, however, I find I'm playing devil's advocate with myself. Those who compete at pool regularly with little chance of showing a profit over expense in a given year, are, in my assessment, hobbyists paying for either the joy or privilege of competing.
Pro pool is more robust than it has been in quite some time and more players can make financial ends meet than in many years. Nonetheless, outside of those focusing on the scraps available to them in regional action, I don't think many players under Fargo 770 are capable of having earnings that exceed their participation expenses.
My fellow AZB poster JAM has done the best at pointing this out, but she's right --- despite pro pool's growth, undeniable in most respects due to the fine efforts of Matchroom, CSI/Predator and some other event producers, pool is still a lucrative career for disappointingly few.
So then, what's a pro pool player? I know I'll change my mind down the road, but for now it's a player who either makes their primary income from competition or has a reasonable expectation of doing so in the future. For me, that means either Fargo 770+ or someone who can reasonably expect to reach that level (and there are many such players).