Your mother sounds like mine was. And whilst I didn't understand just how toxic she was until I was recovering from a relationship with a sociopath, I always knew I would parent my own child completely differently to how she parented. I vowed it to myself in my teens, and I kept that vow.
The thing is, she may turn into a doting grandmother, but your child will see how she treats you - and most likely won't like it. My son once said as we were driving home from my mother's, " I don't like nanny'. 'Why is that, sweetheart?' I asked. 'I don't like the way she treats you.'
They don't realise it, but their actions speak volumes about them, and say nothing about us. People who have been brought up with love see it and judge them, just like your DH does, just like your friend did and just like my son did.
She never will give you her approval - her own feelings of superiority and even acceptableness depend upon her being better than other people, especially her daughter. Because she probably feels extremely threatened by you and your happy life. Other people aren't judging you by her twisted standards, so please don't think her standards are worth anything.
My mother had a horrible childhood too, and so did her grandparents (who brought her up), and probably so on and so on ad infinitum. Personally I don't see the point in casting blame, because where does it get anyone? But our gift is that the genetic line of abuse ends here, with us.
You are a better person than her, because you keep a nasty woman in your life in spite of her actions. She can't even find a kind word for her own daughter, who deserves to be loved and cherished just like you cherish your own DD. Instead, she sucks the life out of you, because you are looking for validation from someone who gets their kicks out of invalidating you.
Your DH is right. She is not worth a moment of your worry. The book suggestions from PPs are really good ones, and I would add 'How to Survive in Spite of Your Parents' by Margaret Reinhold.
But always remember - it's her, not you.