The Bnet version of Warcraft 2 came out after Starcraft.
Again tho, Warcraft 2, Red Alert, etc were all only 1 on 1. Something to be said for that step, but Starcraft was somethign else.
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I played team games in C&C and Red Alert on Westwood Studios Westwood Chat/Matchmaking service.
In any case, Quake demolishes SC for its contribution to online gaming. It came out a year before Starcraft, and anyone lucky enough to have access to a T1 line, or even dual ISDNs would host a server. It resulted in QuakeSpy being created, which ultimately evolved into GameSpy, and spawned the Team Fortress mod that laid the foundation of online, class-based team gameplay.
The thing Starcraft has going for it, is that it's arguably still the best and most balanced RTS out there. Not bad for an 8.5 year old game. Kudos to it for having 3 separate factions, that weren't just carbon copies of each other (I hated WC2 for that), with distinctive gameplay styles.
But Quake dominated in multiplayer.