I don't have a problem with the names people call their parents, although I agree that, regardless of the names I'd call them to their face or within the family, I'd probably be inclined to to use a more generic "Mum" or Mother" when referring to them elsewhere.
What I found really odd when my children were little and babies was the unusual grammar adopted by some parents and hospital staff. When first born my children were always called "baby" by maternity staff, never "the baby" or "your baby". I never understood this loss of a pronoun. I suspect it stems from not remembering names, and babies without names, but it struck me as a unsual. I'd be musing on this anomaly and often missed what they were actually telling me.
Then as toddlers there are the parents who refer to themselves always in the third person. As in "Nigel give those scissors to Mummy!" When they mean " give them to me" Grandparents sometimes do it to. I can kind of understand it with tinies, that it's done to avoid confusion, but even little ones soon grasp the concept of "Me" and understand that Mummy is also a "Me" when she's talking about herself. Then there are parents who don't shed this habit for years, still using the third person when their kids are 10+ years old.
My own family had so many family words and phrases it's a wonder I can speak English at all!