The aircraft always has third party liability insurance as far as I'm aware that covers passengers so long as instruction isn't excluded you will be fine. Employers Liability insurance normally only covers the landside part and the airside part would normally be covered on the aircraft insurance policy.
Theoretically, the aircraft insurer, or ambulance chasing scum (no win no fee) could attempt to take an instructor to court suggesting that the instructor was in some way negligent. It's unlikely but it can happen. In this instance it would kick off instructors liability insurance, but honestly this would only happen when A) the instructor was worth any money at all (LOL!)
B) The instructor had liability insurance to actually pay out
C) the ambulance chasing scum were just idiots.
Basically in my opinion if you have such insurance you're more likely to get sued and would be feeding the fire. You make your own choice, I didn't and still won't.