I don't recall ever playing BattleTris, but Tetrinet is one that got passed around IRC a lot in the 90s. It was a blast. Seconding the notion that their gameplay is a good source of inspiration.
TetriNET was fantastic. My students played SO much of it in class after they finished their assignments for the day. We even made/used custom skins to improve visibility and thus gameplay
https://en.wikipedia.org/wiki/TetriNET