I had a rough birth with my son, and contacting grandparents to tell them about the birth, wasn't at the forefront of my mind. If your complications meant that either you or your baby, could be in danger, I'm not surprised your husband didn't text his Mum. Not only did he not have time, but very likely, he didn't have the headspace. His priority, and rightly so, was you and his child, not his Mum!
You told both sets of parents at the same time, it's not like yours knew before his. Most grandparents would be understanding in the circumstances...not your MIL though. Somehow, she's managed to make the birth of her grandchild, all about what she wants/feels. As for turning up unannounced, an hour after you got home, that was rude. Again, most grandparents would have respected that you needed some space.
Unfortunately, it sounds like your MIL is very domineering and is used to having her own way. When challenged, she turns on the water works, and makes out she's the victim. In other words, she's a conniving and manipulative.
Your MIL isn't going to change, in fact she'll probably get worse as she ages.The only way to get through this is for you and your husband to be united. You need to use the 'grey rock' treatment.
Your MIL's comment is extremely odd. She's clearly someone who likes to be centre of attention, and thrives on drama. Make sure your MIL is never on her own with your child.