The Game and the size of the Table are the two factors for me that define it as Billiards or Pool. Seven foot pocketed tables are Pool Tables use for Nine-ball Eight-ball, Ten, and Seven-ball and a few other quick games.
Where as Snooker, Three Cushion and Straight pool and One Pocket are generally played on nine and ten foot tables depending where in the world you are. Snooker and Three Cushion are played on ten foot tables I believe.
Along with where you are playing as well as who you are playing defines the game. As either Or. Pool was a coin phrase I believe back in the Late Forties and Fifties from service men who learn of the games being played over seas and brought them home with them. A fast game where a quick buck could be made.
Most Billiards Tables were in or near of a Race track betting parlor where bets on Horses were Pooled together. So some how the term Pool stuck to the Game of Pocket Billiards. I don't Know when or why only How.