LEOSA relies on the definition of a "firearm" from 921;

(3) The term “firearm” means (A) any weapon (including a starter gun) which will or is designed to or may readily be converted to expel a projectile by the action of an explosive; (B) the frame or receiver of any such weapon; (C) any firearm muffler or firearm silencer; or (D) any destructive device. Such term does not include an antique firearm.

From the NRA's FAQ section on LEOSA;

1. Does the agency who qualifies me need to make a record of the make, model, caliber, or serial number of the firearm I qualify with?No. LEOSA does not require the agency to maintain this information. This is a frequent concern given the statute's use of the term "type of firearm." LEOSA authorizes the carrying of a "concealed firearm" of the same "type" the individual receives certification for. As there is no case law interpreting this wording, the word "type" should be read to conform with the dictionary definition; something distinguishable as a variety. Accordingly, "type" of firearm should be read to mean either long gun or hand gun, which would permit you to carry any type of legal long gun or hand gun based on your qualification and not one particular make, model, or caliber. As an action outside of LEOSA requirements, the creation and maintenance of a database may expose the agency to liability, as discussed below.