While it could seem like it's lazy research, they probably do not want to mess with placing characters from other Final Fantasies' canon into the world of XIV. It's more likely just an homage in appearance, but using the name of the lore-irrelevant version of the boss.
As others have mentioned, the feature of the Gambit system is that you had the choice to automate your party's actions based on commands you programmed. This is what the Gambit menu looked like:
The lower on the list an action is, the lower priority it is for the character. As you can see, the first action a character will use above all else is for if a party member is dead, and so they will use "Phoenix Down" on any character. The game is smart enough to only use it whenever someone is dead so you don't need to specify further. After that, it's to Poisona. Then if all characters' HP are above 70%, it'll go to the next action, which is to cast Aero on any flying-type enemies. If there are none or they are defeated, the character can finally use the basic Attack function on the next closest enemy. I hope that helps.