Office Pest, something that hasn't really come across so far on the thread is the difference between the state (Urgency/Distress) and response (ATC calling out emergency services etc).
Of the former, there are three I guess:
Mayday/Distress
Pan/Urgency
...and what might be called 'none of the above/ops normal'.
This is used by ATC to determine, and by flight crews to demand, a certain level of priority.
ATC react to a situation by declaring an emergency category, which might be one of;
Full Emergency
Local Standby
Aircraft Ground Incident
Aircraft Accident
Act of Agression
....there are more than that, but that gives you a flavour. Also, each cateogry might well be sub-divided depending on the situation.
So, a 777 might suffer an engine failure, and might declare a PAN.
My ATC procedures mandate that I have to declare a 'Full Emergency' and the AFS meet the aircraft.
A 747 might suffer what is in effect a minor hydraulic failure, and not actually declare anything , but again my ATC procedures mandate I have to declare a 'Full Emergency' again.
Another aircraft might declare a PAN due to a sick passenger, and as ATC I do nothing more than allowing it to 'jump the queue' and land ASAP.