They were Strike Suit Zero and Goat Simulator. Both run perfectly outside of Steam, but neither one of them will save without being logged in; Strike Suit Zero even outright tells you when you start up that Steam isn't running, so the save file can't be loaded. I haven't tried the new Director's Cut, though.
Honestly, that's just how some games work. Anna is the same way: It requires the CD key and account login information in Windows, and absolutely nothing in Linux.
I have Strike Suit Zero on Steam. Ain't played it yet.