When I was planning to go to HAI I asked that same question of them. Their answer was that you are only allowed to work legally as a flight instructor on the J1. (But agreed, the definition itself doesn't seem to rule anything else out, since it states you're allowed to work using the professional qualification you trained for, which it could be argued is CPL, not CFI.)
Not only that but I believe the flight school administering the J1 also has responsibility for you sticking to the terms of the visa during your whole stay.
Si