Which muscles do I need to stretch to improve folding, head-to-knee, and seated fold?

I’d work on your hamstrings. People typically have tight hamstrings and that can result in all sorts of different pain including back pain. So work on those and then also just general back and leg stretches will also help.

Try a Pilates class. I am in solidly ok shape and was amazed at the flexibility and strength it required.
I think that if you were to do it often, you’d improve core strength and overall flexibility without getting hurt!

It sounds like maybe the back of the thighs through the glutes… That's usually what I feel is limiting my stretching in those ways

Maybe side to side back muscles, stretching up and over. Backs of legs too. And hip but I'm not totally sure how. Lunges?