Part of the trouble with being a parent, it always seems, is that when your children are grown and have their own, you naturally want to step into the parent role again. After all, these are people who you brought up. Of course you know more about parenting than them! Of course you know best! Ad nauseam.
It's harder when the to-be parents make different parenting choices. Anyone with the tiniest crack of insecurity (read: everyone) can start to take it as a personal criticism. Which is silly. There is no single parenting formula or rule that works for all children (except, perhaps, things like "don't throw babies out the window"). Let the baby play and have fun at daycare - oh no you are robbing them of parental affection and wasting money! Let them stay home with you on a reduced income - oh no you don't care about your child enough to want to afford a better lifestyle for them, and they'll be socially stunted! Honestly, it doesn't matter. As long as the child knows they're loved, they'll be fine, so do what makes you most comfortable and happy. A happy parent is better able to care for their baby, after all.
I think you handled your mother fine. Were I in your place, I would be very hesitant to have her spend time alone with the baby, at least for the first few months. This is because even thought she backtracked on the phone when caught out, she still said what she did and it will still be her opinion, even if she doesn't voice it as much. Although I think you can expect more from her - she'll test you to see just how far she can push. I also wouldn't have your sister nanny, because you don't need another person on your mother's leash around unless there are no other choices (and there are for you), plus it will emphasize to your mother that she was right and that it is appropriate for her to be making parenting decisions for you.
You need to be very firm with your parenting decisions. They are not hers to make, and if she takes umbrage with that, it's her choice to and not your fault. It has nothing to do with what those choices are; you need to nip this in the bud, or it will spill over into other things. You are the parent here, not her. I wouldn't even explain your reasons for your decisions at all, since that opens it up for discussion. Just say "we've decided that doing XYZ works for our family and it's not open for discussion. Now, about that delicious bean dip I've been meaning to tell you about..."