rich_g85,
There is a distinction between the level of distress (i.e. is it PAN or MAYDAY?) which then dictates the level of priority ATC give the aircraft, and the actual problem that has been encountered.
You might have an inbound aircraft with a sick passenger, the level of distress going all the way from;
- Nothing abnormal, just need an ambulance to meet the aircraft at the gate
- Medical Emergency
- PAN Medical Emergency
- MAYDAY Medical Emergency
All ATC do is prioritise the aircraft as appropriate. The 'response' level would be similar regardless of the level of distress; request an ambulance on stand and pass the pertinent information.
Likewise, a modern airliner may have suffered a slight hydraulic fault in one of its many hydraulic systems which in fact have absolutely no impact on the operation of the aircraft, therefore is given no level of distress by the crew, and yet ATC in the UK will probably respond with a 'Full Emergency' as that is what the ATC Manual states will be the response for any hydraulic failures. The semantics can cause confusion, as you can see the flight crew have not 'declared an emergency', yet ATC have initiated an 'emergency' response.
Levels of response such as Full Emergency, Local Standby, Aircraft Accident Imminent, Aircraft Ground Incident and so on will be defined in each airport's emergency orders in terms of what triggers each level and what exactly happens on the part of each agency.