WOW. If she can't respect your feelings about how YOUR child is raised, then maybe she isn't the best choice to babysit.
You said you corrected your child, but have you corrected the grandmother? Have you made it very clear how serious you are about your feelings on this? Because she is really crossing the line, and you have every right to be mad.
With that said, you said you were "thrilled" before. Is everything else great? Is this worth getting a different babysitter over...?
And one more idea. I don't know what your relationship is with your MIL, but maybe you could sit down and tell her how lucky she is to be a grandma and that she should be happy with her new role and embrace the new name... Maybe that's it! Is she young-ish? Do you think she's uncomfortable being a "grandma" and is trying to stay "mom" instead...? If so, maybe she needs a different name than grandma or memom. Something younger, cooler...? It may just be her insecurities screaming, and not meant as a slight to you at all.
Those are my thoughts! GOOD LUCK!