What ability is this, and what class uses it?

[Abriael]

Take apart the events in a cutscene and you have basic things like: interesting skills with a highly visual animation, battle system where positioning matters, fast response times.

What makes them spectacular isn't only the basic animation, but the direction, the camera angles, the music, the linking between those animations. It's not really possible to do that outside of a directed cutscene, besides with quicktime events.

They actually did something kinda akin to that with FInal Fantasy XIII, but to achieve that they had to remove part of the control from the player.

The more specific an animation is, the less adaptable it is to the environment and to the other mobiles. This means that the most spectacular animations simply cannot be in player control, because the player needs to be able to use all his tools in every area of the game.