You are correct in your statement that there is only one Class A license; however conditions or restrictions may be imposed.
Here is the pertinent language.
Massachusetts General Laws Chapter 140, Section 131 deals with firearms licensing:
Licenses to carry firearms; Class A and B; Conditions and Restrictions
All licenses to carry firearms shall be designated Class A or Class B, and the issuance and possession of any such license shall be subject to the following conditions and restrictions:
(a) A Class A License shall entitle a holder thereof to purchase, rent, lease, borrow and carry; (i) firearms, including large capacity firearms, and feeding devices and ammunition therefor, for all lawful purposes, subject to such restrictions relative to the possession,use or carrying of firearms as the licensing authority deems proper; (it repeats the language as to rifles)
The statute does state that a violation may result in fines, revocation etc., but the provisions of MGL Chapter 269, section 10 shall not apply. That statute is the "dangerous weapons statute" (same as outlawed martial arts weapons) and 2 and one half years in jail.
The licensing authority has the authority to place conditions and restrictions on the Class A license as it deems proper.
One could argue that the limitations were arbitrary or capricious and defeat the purpose of the Class A license, but the target & sport provisions on a Class A license have been with us for eons; you may recall under the old statute you always asked that the language "For protection of life and property" be endorsed on the license in order to clarify this point.
I hope this answers your questions.