I don't know why these matters aren't discussed in advance so any problems can be smoothed out asap. I asked what my S-in-L's mother wanted to be called and, as she was already a granny to her daughter's children, then that was settled. I didn't want to be called granny anyway as I think it's aging so was happy to be nan or nanny. Her husband was granddad already so my ex husband was asked to choose something else and opted for grandpa and his second wife is referred to by her name.
This has reminded me that, now my second daughter is pregnant, I must ask her to see if her partner's mum minds if I am nanny to their child. She already has grandchildren so, if she's already a nan with them, there may have to be a second word added on e.g, Nanny first name. The other grandparents don't/won't see my grandchildren that often as they live some distance away and are already closer to their present children so probably it's easier for me to be just nanny but it can all be discussed amicably.
I'm not keen on this nanny 1, nanny 2 business because it makes it sound like one is a second class citizen but I believe it is important for grandchildren to have a good relationship with all their relatives and, unless I'm especially asked over, I keep out of the way when the other grandparents visit, to let them have quality time as I am able to see my grandchildren several times a week.
There's no need for jealousy; each person can be valued for different things. One side gives expensive presents without asking what the children or parents want, I ask for one main suggestion of what they want ( this may be expensive or otherwise but is what the child has asked for) and then I give a few cheaper surprises or things I know they need in the clothes line.
I do think our adult children should have some consideration for the grandparents' feelings re names, visits etc. but it could depend on how they have been bought up/who is the dominant partner. I will never have a daughter in law so would have encouraged my daughters to give their fair share to the 'other side' unless there was some serious problem but they do that anyway without any encouragement.
For those with harsh d-in-laws with sons, maybe they need reminding that their child may grow up to copy their behaviour towards the maternal mother in law!!!