I would classify a professional pool player as having 5 qualifications:
1. A person who is a student of the game. By student I mean they learn the history of the game, they learn the ins and outs of exactly how balls respond to english, english after hitting a rail, how to cheat pockets, jump effectively, etc. These folks are not one trick ponies and their knowledge of the game is much deeper than an amateur.
2. Hone that knowledge through hours of time devoted to practicing, drills, seeking out opponents to play, and tournaments on a daily basis. A professional does not simply sit on the knowledge they have. They are constantly looking to be tested, apply, and pick up new knowledge.
3. A burning desire to win no matter who the opponent is. This one is a more of a personal preference than an absolute requirement. To my mind, pool and all its forms is not simply a game to professionals. It stings badly when they lose, and not just monetarily. A true professional doesn't feel sorry for their opponent, doesn't care how much money they take from someone, at least in the moment of play. To quote Earl Strickland, they will "shoot your liver out and hand it to you".
4. A big win doesn't curb the hunger. A true professional pool player doesn't care if they just won the U.S. Open 9 ball championship last week. Within a week, they are ready to go to small town nowhere and win a $500 first place prize. This doesn't always mean they will, because a pro will take a great deal of time to plan what tournaments and events to attend around their practice schedule, and is also likely to fit some money matches in as well. Yet no matter how often they win, the hunger remains. When the hunger dies, you will often see these folks retire into the sunset.
5. The money is secondary to the game itself. This may be being a bit of an idealist, but here me out. There is no doubt these folks want to make money, and they enter big events because of big pots. But I believe at the end of the day, a true professional wants not only the money, but the satisfaction of knowing they outlasted a field of people who are on fire for the game as much as they are, and then took em all down. These folks also are highly respectful of the rules and stand for both integrity and respect. You can disrespect them (sometimes to your peril). But if you cheat, disrespect the game, or don't give your best, deep down they are going to be more ticked than you can imagine.
The money does matter. But being a professional is a combination of skillset and mindset. If you really think about it, it isn't the money that a pro make that keeps them from competing in amateur events realistically. That is just the visible measuring stick of their success and attitude. What really matters is the level of play, the dedicating and time spent in the game, and how they approach a match. A professional isn't having fun with mates at the table. They're a cold blooded assassin ready to gut anyone who steps up to challenge them. This translates into more work on honing their knowledge and abilities, which most players simply cannot match.
As for a pro event, I would classify that as any event where the entry fee is over $100. I know some pros may laugh at me, and some amateurs. But really think about it. If you are willing to plop down a hundred dollars on a tournament or a match like its nothing, you either are one of the most arrogant aholes on the planet, or you know if your heart of hearts you can either win it back later or you're gonna gut the person who takes your challenge up. For a lot of folks, $100 means something, unless someone else is putting the money up for you. Most folks don't have backers and won't bet their own money past $20 a game (thank you Poolhall Junkies). A pro, ready to lay money on the line for a big payout and a chance to prove they are number one for the day/week. That means a pro event is going to raise the stakes to attract the highest level of player and competition, not just for the money, but the glory.