Down here in the South we would just call that a "cousin". At least when I was growing up, all confusingly related people or not-really-related people who are neverltheless treated by the second and third generation as if they are family, were given the blanket designation of cousin.
You just made me aware of something.....I'm in the South too and I call the wives of my first cousins "cousin". The mother in this post is more distantly related through marriage and DH has never considered her of any relation to him. May have to do with his feelings towards her- he does not like her and her husband, but to his credit he is always polite to them at Christmas (the only time we see them). He has made it clear to me in private that he wants nothing more to do with the mother or her family than what's "required" at Christmas. Probably why I don't think of her as a "cousin" either.
OP, I'm curious if she was so horrid about any of the gifts from others. Did you notice her commenting about all the gifts? It might be good to try to think if anyone gave her kids something she didn't complain about and find something similar.
Now that I remember, she made some snide comments about another person's gift a few years ago when it was something crafty. Guess no messy, crafty gifts for these kids from now on. I had given books to the girls in the past and that went over okay, but as the kids get older I feel it is harder to pick out a book they would like if I don't know them. They were also given necklaces without a fuss from mom, so I may remember something like that for next year.
I have also given play-dough to other kids. In my experience, every kid loves play-dough and it always seemed to be a great gift. It also assists with working on fine motor and sensory skills which are essential for a child's development. I played with it often as a kid and I lived in a house with beige carpets. The rule was that I could only play with it in the kitchen at the table where there were tiled floors. My mother made it very clear that it wasn't allowed anywhere else in the house. So when I give play-dough or other crafty gifts, I just assumed that the parents will set similar limits for their kids to avoid messes. But every family is different and from now on I will try to remember to ask the parent first before giving these types of gifts.