Part of the reason I still asked this question (even though it has been discussed previously on this site) is that, although there generally seems to be agreement that extra-curricular activities are included in direct expenses, there are also disagreements about this, which is very clear from the response I am getting here.
I'd love to see data on what judges are ruling about this. I've heard of judges ruling all sorts of different ways but some of that is also coming from the Missouri courts. For anyone else reading this that has my same question, my partner recently went to mediation and during that, the mediator (This is a Jo-co court appointed mediator) offered to explain to his ex "how direct expenses works". She stated that extracurricular activities ARE included in direct expense, and, since he is paying her child support, technically she is responsible for all of the costs. Our lawyer has confirmed the same. When my SD asked to have dance class twice a week, my partner asked our lawyer what he should do and he suggested signing her up, assuming Mom would agree to it and ask her to pay for it. This is not how my partner proceeded, but it's what the lawyer suggested.
We have not gone to court about this but are hoping things will resolve in mediation. All of Mom's inappropriate actions have been clearly motivated by money so hopefully when faced with the reality of court fees, lawyer fees, a guardian for the kids, and possibly ending up paying support to my partner, she will be willing to compromise.