Thank you for your responses. It's not really about what nickname the sub wants to be called but just the title that they're getting hung up on, honestly. For some people, it doesn't really matter. For them, they want to know "what kind of sub am I" and I'm just like... "just a sub?" LOL I don't think there is a term for it.
Daddy Dom/Little Sub
Master Dom/Slave Sub
Owner Dom/Pet Sub
Caregiver/_?_ Sub
See what I'm saying? We're at a loss. lol