Not sure about other countries, but New Zealand allows the pilot to nominate the desired response, in accordance with the aerodrome emergency plan. If the pilot does not nominate a response level, ATC will activate an appropriate level.
When an emergency occurs in flight and adequate communications
exist, the pilot is responsible for advising the ATS unit accordingly and for
nominating the desired state of readiness of the aerodrome emergency
services. If adequate communications with the aircraft do not exist, the
ATS unit will assess the situation and bring the aerodrome emergency
services to the state of readiness considered appropriate.
Reference AIP NZ ENR 1.15 7.2.
For example, a pilot may declare a MAYDAY and request local standby. Essentially the MAYDAY is the priority required from ATC, and the local standby is the response required from the aerodrome, rescue fire, and other aerodrome resources. This could be for example an engine failure, although the pilot reasonably anticipates a safe landing will be made.