I wouldn't call them professional gamers, but I'd call them professional entertainers. Because that's what they do, they don't really get paid money for the games they actually play, they get paid because they entertain in the process. That goes the same for every Youtube gaming channel, would PewDiePie be as big as he is if he just played the games with no commentary, no webcam, no anything? Of course not. Professional gamers, on the other hand, are paid purely for playing the game, we don't get any kind of commentaries by them during it, they just play, win, or lose.
And you're missing out that many pro players are hired by companies. Companies like Fnatic give their players salaries across all their games, they are paid professionals in every definition of the word.