DPS is used in FFXI, just not as a synecdoche
as it is in other games. I suspect this has to do with the speccing system used in other games, where a particular character can only have on set of specs set on it, so you spec it out for maximum DPS and just call it that from then on. As in, "my DPS guy." Then it's shortened to "my DPS" because everyone knows what you're talking about.
This is true to WoW.
When each Class as multiple specs, DPS just became the short hand for Damage, or Damage Dealer (I think it is labeled as Damage, too lazy to log in and look). It was the evolution of terms already out in the MMO/Mud world.
Heck it goes for Tank as well..
Tank? I've never outside of talking about a Game used Tank as the way we use it.
"Oh hey, I'm a tank"
*Person not knowing what you just said, pictures your face on the outside of a tank