When she came up with the name she was musing that she wasn't a "Kitten" anymore--Storm's nickname or her--but a "cat." All grown up. As 14 year olds tend to think they are. Especially those that have been to the far side of the Universe and helped save the world.
She's never been a Kat always a Kitty. In the 30 years later Days of Future Past story she was Kate.