I think the pouncing style animation shows up when you get the Choco/Mog Summon Materia. (The Chocobo's on the ranch dance, and also pounce)
And, I think the Seph's hair does stand on end like that, when he's entering the battle. (He flys down, and his hair glides down also)
So the animations are in the game, it's just that you're not observant enough to spot them.
Anyway, I want to ask something that's un-related to this topic, do you know where the Independant Materia (I can't remember the name of it) on the side of the crater is? (When you're climbing up it)
If you do, could you please tell me, it's the only Materia in the game that I have never picked up, in all of the times I have completed it.