You very much need to step completely away from this friendship. Even if she meant well, which she most likely did, her way of interacting is NOT compatible with yours (she’s wound me up too!).
Honestly, just because you both have same aged babies, doesn’t mean you need to be best buddies.
Recall a similar relationship when I had a new born, and wish I’d cut ties muuuuch sooner than I did. She made me second guess almost all of my actions, even the health of my baby, and cast a small dark shadow over the period. In her case, her narrative was basically everything she did was supersonic new age… and anything that wavered from her line was Bad and Dangerous. so e.g. only cloth nappies, no plastic toys, we both breastfed, but had I formula-fed she’d no doubt have made me feel dreadful for bottle feeding.
She used to lecture endlessly. I liked some, if not all of what she stood for, it was the STYLE of her interaction that didn’t work. She wasn’t empathetic or actually very friendly / thoughtful at all. Your scenario reminds me of this, even if a different example. wtf is she contacting your partner, and after 12 hours?! Good for you for not being glued to your phone, also!
So yes, detach totally, or give her another chance, and if she repeats this sort of annoying behaviour then let that be the end of it all. That all said, if she was great in every other way, you’d probs overlook this, so Im guessing she’s irritating in other ways too!