I agree the MIL is out of order here, but can I just make a comment.
ds is a red head. When he was a baby he couldn't be in the sun at all ever, he was constantly covered in factor 50, under an umbrella, with a sun hat on etc. If I ever forgot, or we walked somewhere and got the sun, he would start to colour up.
I had to be really vigilant.
But my brothers and I are not red heads. In fact we can all tolerate a lot of sun before we would have a problem, and it was the same as kids, my mum has zero experience of a baby like ds.
I can remember being out with ds and a friend, then about 1 year old. She didn't have an umbrella and her dd of the same age was in the sun a lot, I felt anxious (and I am seriously not an anxious person) just watching her. At the end of the day, her dd had a light tan line, my ds would have been lobster red.
It wasn't until I had dd whose skin is like mine, that I realised that most kids aren't as sensitive as ds, and that I didn't have to be so cautious with dd.
If my mum had had ds in the early days, and they had been sitting in the garden, I can well imagine a scenario where he would get burnt. My mum is great, and she has always been serious about following our wishes with our kids, even when she thinks we are batshit, she has a very strong sense of them being OUR children and it is up to us, but If I hadn't spelt it out, ds would have got burnt with her. Dd on the other hand would never have got burnt.
So while I would be cross, I can imagine how it happened. The issue for me would be her response, and her willingness to listen in future.