MIL has a longer history with your husband than you have. She had control of him when he was a child and did some brainwashing back then that he has never gotten over.

It's your job to explain to daughter that we apologize, even when something is just an accident. Would have been nice if grandma and grea-tgrandma could have helped with framing it that way, but they're dopes, so it falls on you.

It's good that great-grandmother apologized for name calling. This may be kind of working out in a way that will be a learning experience for your child. Talk to her about it. If she did apologize, praise her for that. She can come out of this as the bigger person. If your husband organized the exchange of apologies, that's really okay too. He's a parent and he can do that.

The most important person in all this is your daughter and how she comes through this whole tempest-in-a-teapot. Grandma and great-grandma are not blood to you, but they are to your daughter. So, maybe, let her lead the way a bit in this. See what's in the email, when it arrives. If MIL makes some attempt for taking responsibility for making a mountain out of a molehill, then maybe let life move on.