My MIL was Nonna (and late FIL Nono) already when I married DH - due to older brother with DC. No Italian connection that I know of, although French speaking Egyptian, not English. I therefore think that Nonna must be used more widely than just Italy.
My DM didn't want to be Granny either, because my Granny (her mother) died less than a month after I had my first DC, and I think it felt like a usurpation. She therefore settled on Grandma (Nonna was out of the question, given my MIL).
But in answer to the question, I absolutely agree it should be your call. Yes how you parent the child is the responsibility of the parents, and the parents only, but that shouldn't give them the right to define who you are, even in relation to their children. They don't get to tell the teachers at nursery, no you can't be called Auntie (First name) but must be called Mrs (Surname) (Auntie (First name) was the rule in my DCs nursery, and I always found it a bit cutsy and weird, but the nursery had the right to decide that is what they wanted). I agree with you that it does seem controlling to foist on you a name you dislike and thereby attempt to make you into someone that you are not - whatever your reason for not wanting to be Granny. A name is about who you are, not about who the DC are. And just as it is inappropriate for a grandparent to try and insist on their favourite name for their grandchild, parents get naming rights of the DC, until the DC grow up, when they are fully entitled to change it by deed poll if that is what they want to do, it seems inappropriate for the parents to insist on their favourite name for the grandparents.