We must not get confused between the degree of urgency applied by the flight crew (MAYDAY/PAN) with the emergency response declared by the airport (usually ATC).
They are not directly correlated.
A mayday in the cockpit might not elicit any emergency response by ATC. A small issue in the cockpit that doesn't warrant a PAN might actually cause ATC to initiate a Full Emergency
Once an emergency response has been initiated by the airport, usually it can only be cancelled by the Airport Fire Service.
In the airport's Emergency Orders, each category of response (Local Stand By, Aircraft Ground Incident, Full Emeregency etc) will have a fixed response level by outside agencies. There is no discretion on the part of the emergency services to dispatch what they think is appropriate.