So, Nursery Rhyme is pretty confusing re: names, so here's a short list of what the heck's going on when she gets called a bunch of different things.

The basic breakdown is like this:

Official name: Nursery Rhyme
Alias (used rarely): Caster ("Enemy Caster" is used OOCly to differentiate her from Playable Caster, a different character which uses the same title in her canon.)
Calls herself: Alice, which is also the name of her closest friend.
Codename: Cheshire Cat, applicable only to The Xavier Institute.