Where does it say "hunting, target shooting or self-defense"?
In NH's statute:
http://www.gencourt.state.nh.us/rsa/.../159/159-6.htm
I agree the safe answer is "all proper purposes" based on this conversation.
I disagree that the application will get denied and you will lose $100 if you put "self-defense". Self defense is literally one of the examples of "proper purposes" spelled-out in NH's statute governing the issuance of licenses. If someone processing the application doesn't know their own State's law and kicks it back, I'd do as they ask while supplying them some educational materials for their consideration. But, in any case, I don't think you will lose $100 if you write anything other than "PROTECTION" or "ALL PROPER PURPOSES". If those are the only acceptable answers, they should just fill that section out themselves…