Most of the forms keep their name, but Zygarde Perfect Forme is to be called Zygarde Complete Forme in the west. It also describes Ash-Greninja as the form that Greninja takes when the bond between it and Ash is raised to the limit.
Here I will 列表 off all the 神奇宝贝 Ash caught within the first 16 seasons of the anime, what season he caught them, what episode he caught them, where he caught them, and their original trainers (if they had a trainer before Ash). Here are the first 20.
Kanto (Original Series)
1. 皮卡丘 (episode 1- Pokémon- I Choose You!)
Pikachu was Ash's starter 神奇宝贝 that he received from Professor Oak. He received 皮卡丘 because he woke up late on the 日 he was supposed to start his 神奇宝贝 training. 皮卡丘 didn't like Ash AT ALL when Ash first got him, and was very disobedient, refusing to help...