Possibly a clue to the priorities of the different Fire Services lies in their titles, particularly the order in which the words fire and rescue appear. The title given to airport fire services is Rescue and Fire Fighting Services (RFFS) whilst local fire services either have the title Fire Brigade or Fire and Rescue Service.
As an aerodrome FISO my reaction to a MAYDAY or a PAN would be to categorise the “problem” using a list very similar to Gonzo’s and then follow the procedures laid down for that category of emergency. Some of these procedures involve only the airport RFFS whilst others also involve local emergency services