The reverse is generally true too: if the squad had First Fired its inherent FP, it could fire the LMG alone as First Fire (on the next expended MF). But if it would rather combine its inherent FP plus the LMG,
both would be halved and marked Final Fire. There are some additional restrictions and requirements in this situation (including Sustained Fire), but that only gets us deeper into the weeds...
And while I agree that A8.3 is dense, you might have missed all this if you are using an early copy of the ASLRB. A final sentence was added to A8.3 which answers your question succinctly:
If a unit, or any SW/Gun it possesses, uses Subsequent First Fire (or Intensive Fire) then that unit and all its SW/Guns are marked with a Final Fire counter.