She maintains I'm overreacting
-Tell her that she can tell herself that all she likes if it makes her feel better about what happened, but if she's got any sense she will realise that the important thing is that this is about YOUR life, and YOUR children. So actually, even if you were totally overreacting, YOU still have the whip hand. If you feel, for whatever reason, that the dynamic doesn't work for you and YOUR family - you'll end up distancing yourself, and she will suffer - not you. You have no obligation to put up with anyone else barging in and telling you how to care for your kids, clean your home, do your job, fold your knickers. Even if you do all of those things extremely badly - it's still none of her business, and you can decide anytime you like to tell her to stay out of your home and life if she can't completely accept that you're just as much an adult as her next-door neighbour and she has no more right to tell you what to do than she does them.
she's just so adamant she was right
Well, she wasn't, was she? Firstly she wasn't right because it wasn't her child and no she did not have the slightest bit of 'responsibility' - as per above. And secondly, she wasn't right because the doctor confirmed that you were right and she was WRONG. Again, she can tell herself she was right all she likes - but it will only convince you even more that she is perhaps someone you don't want so closely involved in you and your kids life and decisions, as it's clearly only gonig to cause upset.
So - her choice. She can open her eyes and realise what relationship she actually has here (clue: it doesn't involve any kind of parental role, at all) or she can fuck it up and end up being the granny who doesn't get to hear about/be involved in stuff because it's not worth the hassle.