I guess I don't. Does post-training from another (larger) model not fall under the umbrella of distillation? I'd imagine it leads to the same spiky-ness issues...?

Distilling you don't have the actual model weights of the teacher. All you have are the teachers answers to a lot of questions. You then teach your own smaller model to answer more similarly to the big teacher model.

Fine tuning you have the actual model weights of the original model, you then train that model to answer in a different (or better) way.