If you complete the HAI JAA CPL(H) course you will receive a JAA licence that is identical to the licence you would receive if you did all your training at a UK based school like Oxford or Cabair. It will make no reference to the country where your training took place.
After you complete your JAA CPL(H) you will then complete the FAA Commercial pilot certificate. The two qualifications are completely independent of each other. The only connection is that you are maximizing your training to use the flight hours toward both qualifications to the extent alloed by FAA and JAA ratings.
The other licence you mentioned is for pilots whose JAA qualification is issued on the basis of a foreign certificate.
Enjoy your training!
Ts&Ps