When is an aircraft-carrying submarine a true carrier?

Should every submarine that carried an aircraft count as an aircraft-carrier submarine?

No. Most used aircraft as auxiliary scouts, while only a smaller group was designed primarily to support aviation missions.

Many navies put aircraft aboard submarines to extend reconnaissance, surveillance, and target-finding range. Those boats differ from designs whose central purpose was carrying and supporting aircraft operations. In the first category, aviation remains an auxiliary that helps the submarine perform its normal mission. The distinction parallels the difference between aircraft carried as auxiliaries on cruisers or battleships and a vessel designed as an aircraft carrier.