As has been said, you were being incredibly unreasonable in wanting your mum to be the first to see the baby - they are both grandparents - but you gracefully acknowledged that, and after all we all have emotions that are less than reasonable sometimes, and what matters is how we deal with them.
You aren't being unreasonable at all in wanting your own mum there when you give birth, and not to feel a bit haunted by someone who, with the best will in the world, isn't that close to you, as you actually labour. Your DH is not the one in labour so his need for "support" is secondary to yours to cocoon during the process. I understand that, as I hated any noise, bustle or external influence at all really as I gave birth. Sounds like your MIL understood, too!
I'm so glad things have worked out well, and hope the birth and time after are lovely for you all. But... as someone who would love a MIL as reasonable as yours sounds, and a GM who actually gave a stuff about the childen (or, indeed, her own son, in any genuine manner) please remember her, in the weeks and months ahead? The instinct to turn to your own mum, and to protect the baby, will possibly be really intense, and you need to rein it in for everyone's sake. Your baby needs both grandmothers, as s/he needs both parents, all other things being equal. No baby can have too many sane and nice people to love them.
Congrats on your pregnancy, and I hope you have a lovely birth and post-natal period. 