I actually disagree that it doesn't matter in the grand scheme of things.
The birth of your child is a huge milestone in your life, and nobody else has the right to announce it for you. You had a traumatic birth for other reasons, a lot of things didn't go the way you wanted or expected them to, and you lost a lot of control. This is often the way in childbirth. We hope that things will go a certain way, and when they don't and you find yourself shell shocked and struggling to process what just happened when you thought you'd be happily cuddling your little baby in the picture of childbirth that you had in your head, it can be a lot to cope with.
The manner in which your child's birth is announced to the world is something you absolutely should get to control.
In a world where you don't get to control whether you have an episiotomy or a bad tear, or forceps or an emergency C-section, in a world where you might be given an intervention you don't want or denied the birthing pool or epidural or delayed cord clamping you do want, deciding how, when and to whom you announce your child's birth is something that can and should remain entirely within your control. Your MIL took that away from you. I'm sorry.
The trouble is, at three months postpartum, you might have missed the boat in terms of telling her how you feel. If you'd told her how you felt and she'd been hugely apologetic and said she just didn't think, you'd hopefully be able to trust her not to do the same thing again if you have another baby. If you'd told her how you felt and she'd been unrepentant and tried to minimise your feelings, you'd know she couldn't be trusted a second time round. But because you didn't address it with her at the time, I think you need to use your judgement based on what you know about her as a person.
If you think it was just a silly lapse in judgement, then if you have another baby you and your husband can tell her in advance that this time you would like to be the ones to announce the birth of your child and hopefully she will respect your wishes next time round. If you think she doesn't really care about your feelings and might do the same thing again even if you ask her not to, I'd be inclined not to tell her your child is born (or at least not send her a photo) until you're ready to announce it to the world yourself.
Anyway, what's done is done, unfortunately. But that doesn't mean your feelings aren't valid, and you can use this experience to set better boundaries with your mother in law going forward.
If she posts a lot of stuff on Facebook about your child and you don't want her to, you can raise that with her at any time and maybe mention the birth announcement thing in the context of that conversation.
In terms of getting over it, you might want to get a debrief from the hospital about why your baby's birth went the way it did, and perhaps separately look into some postnatal counselling, if you haven't already.
Your feelings and your mental health as a new mother are incredibly important. Never let anyone tell you otherwise. And you want to be feeling as good as possible so you can enjoy your baby now, rather than looking back with regret.