What is the correct name for an martial arts instructors insurance policy and what is it designed for?

You pretty much have as good a name as any. The name varies by the insurance company that offers it. It is designed to protect the instructor if he is sued for injury to his student or someone that his students may injure. They may also contain sections to protect them if someone gets injured in the school, say a parent trips and falls while bringing their child to class. It would not be a good idea to run a school without some protection.