Hi, Carol!
I'm the mother of three sons, and two grandsons (each from a different son).
Yep, one of the other grandmother's calls herself The Grandma. Same personality as you're describing.
I just finished reading a really funny book for first time grandma's, The Granny Diary. It takes a humorous look at it (and a lot of her amusing advises applies to MIL's too).
However, at the end, she takes a "haha" dig at her son-in-law's mother, as the Auxiliary Grandma. All in good fun, but that has a bite of truth in it (the author has a son, too, she says she knows it's just a matter of time before she's in those shoes).
I'm trying to think back to what the situation was with my sons and their other grandmothers. Hard to compare, as my mother was more into the bottle than into them at the time, so my MIL's were substitute mothers for me and there was never any problem with them and the boys. My boys are much much closer to their paternal grandma's. And, FYI, in their teen years, they loved being with their grandma's!
Jealousy is natural, but you will have your own relationship with your grand babies -- kids have big hearts and they can love a lot of grandparents (thank goodness!)
Mostly I just try not to be around when the Other Grandmother(s) are there (we also have StepGrandma who is very big in one child's life, and Ex-Step Grandma who is fading fast from the other child's life.).