Mahjong also has several variations on the rules, mostly to do with scoring and betting. The Chinese (or Shanghai) rules are significantly simpler than the Riichi (Japanese) rules. Shanghai has you counting what’s in your hand and occasionally drawing flower tiles that do nothing but give you bonus points. Riichi has some minor betting, hand construction, and a few extra rules on when you win a hand.
And yes, Mahjong is pretty similar to rummy; your goal is to draw a tile every turn and then discard one; when you do, other players can take the tile and put it in their hand. Your goal is to build a hand that consists of nothing but 3 of a kinds, Runs of 3, and a single pair. A “Ron” is what you call when you want to take someone’s discard