The change in movement of the rider on some mounts is the result of a bug introduced with 4.1.
It will be fixed in a future patch.
Oh, no. So we're stuck with this for... what weeks? Months?
Do we know which mounts are affected? I know the horses of all types are. What about the cats? Gods, now I'm going to grinding rep on toons that I never planned on just to get rid of the broken animations.
And yes, I'm going to be snarky and say a very sarcastic "thanks" to the person who was supposed to QA that section of the patch code, and clearly failed to do his job. Bravo, whoever you are.
Start enjoying your other mounts I guess.