"For clarification, when I describe a child’s inappropriate behavior, do I stop there? Do I ever need to say or do more?"
When someone else's child is misbehaving you can describe the bad behavior and/or tell them the appropriate behavior, you can tell them what they are communicating to you and others, and what the effects of their communication are.
The child can loose privileges in a class incentive system that has already been set up. The child can't ever loose your love. After all of this, the only other thing to do is to involve their parents in the situation.
If a child is hurting or insulting another child, the parent would want to know.
Just like home, you can't really teach your primary children anything if the Spirit isn't there. Teach you primary children what the Spirit feels like. When you feel the Spirit leave, or people are being mean, stop everything and sing a song and say a prayer. Do not begin your lesson if you don't have the Spirit. If you have to sing and pray five times during a class period to feel the Spirit, that is the best lesson the children could learn anyway. Give them vision. Help them see that certain behaviors drive the Spirit away, and that as servants of God, you can't move forward in life until you feel the peace of the Spirit.
This is the most important thing to teach.
It is harder to parent other people's children, but if you can master your patience with them, then you will be even more prepared to be patient with your own children. If we wouldn't scold another person's child, we shouldn't scold our own children either.