The "eyecon" was first introduced to FFXIV via HW's release, though not every enemy would use it (ARR enemies lacked it). I think it was in 3.1 and then again in 3.4(?) when SE decided to give every enemy that has a gaze attack this "eyecon." SE introduced a lot more indications to the game to make it easier for players to know what they should do (though it always amuses me when a player runs away from the group when they get a "stack on me" AoE).

The first time a player may ever see this eye is in Palace of the Dead, first from the snakes, then peists, then Ahriman (I think?) and then finally Edda (and of course more after). Outside of Palace of the Dead, first interaction with the eye is probably Haukke Manor's Ahriman.