I think the dude in that video was wrong. Some game you use different weapons. But also, you have to make sure that the animal you shoot is a 3 star animal to start with. You can't get a perfect pelt unless you start with an animal that is 3 stars. You can check that by aiming at them and using R1 (RB) to get info on the animal. That info will also tell you the kind of weapon you need to use to get a perfect pelt. Basically you need to get a one hit kill in a critical spot. Usually the head but you need to make sure you aren't using too strong of a weapon.
One interesting thing is that you don't need a perfect kill on legendary animals. On those you can just do whatever it takes to kill them. Shoot as many times as you need to.