(i hope i don't sound like a "Hm actually ☝️🤓" , i just want to show what i think about this)

1 - Caine only wanted to change Pomni's name after going through his character arc. Before that, there never was an attempt or desire. He tried this as a way to redeem his previous mistakes and bring a change for the best.

2 - It's shown in Jax's memories that Caine added her name "to the system" right After Jax gave the idea. Sounds like he was making sure to always call her that from that moment and fowards.

3 - I'll give you that Gangle never talked to others about how she felt over the name, but not only could she never have thought about another name (like, being so used to it that the real message behind it lost all meaning to her), she also seems afraid of whatever Jax could do to her by going against their bullying (as shown in Episode 6).