Once again, the two are not related. What the FAA decides to characterize as a "business purpose" has nothing to do with whether or not the AMA views it as a covered insurance activity. A good analogy would be the situation with aircraft salesmen. They do not need a commercial pilot's license to take a customer for a demo flight. Yet I am fairly certain that their employer has them covered under their insurance policy. So a guy could be flying a non-time demo that the AMA might view as a commercial activity and simultaneously the FAA could view that as still being a hobbyist activity.

This is all supposition since the actual definition of what the FAA decides is a business purpose remains unclear as does the exact times when a non-paid, non-employee sponsored pilot's AMA insurance may not cover him.