YES... Snooker uses 15 reds and 7 colors, generally with a 6 ft x 12 ft table, pockets in corners and sides.
Billiards is a game using NO POCKETS, generally with 3 balls.
Pocket billiards describes any of the games using balls and pockets, including snooker, 9 ball, 8 ball, 14.1, etc. It is a generic term.
Pool generally refers to pocket billiards. The term "pool" came from the early 1900's when pocket billiard tables were installed in the race track betting rooms, called betting pools. The game was so popular in these venues, it became known as "pool" from their "betting pools".
EVER WATCHED A GAME OF POLO?