Very entertaining discussion. A few points from my side of the world, that is Texas;
Whether the pilot says "Pan", "Mayday", or "Blueberry"; if I know the aircraft has lost an engine, ATC will treat that aircraft as an emergency, unless there's a very good reason known to us not to. Obviously, we're going to be taking clues from the crew's requests and tone on the radio as to how much
Immediate assistance is required.
As for the remarks about controllers not knowing much about aircraft performance; while It's obviously been more difficult and expensive to obtain pilot ratings/experience in other parts of the world, I presently work with, or have worked with:
Retired DC-4 thru 727 Captains
Decorated Air Force Pilots from WWII thru Vietnam and later.
Decorated Vietnam era Army Helo pilots
Controllers with type ratings in Lr-25, Hawker 125, etc.
We currently have about six Flight Instructors at my facility, and at least another half dozen with FAA Private/Commercial ratings.
The above list comes from just two medium sized Terminal facilities. Were you to canvas a Center, with 400 some-odd employees, God knows how many former/furloughed airline and corporate pilots you'd find.
And I myself am a CFIIMEI with Corporate experience (albeit many years ago) on the MU-2.
The truth is, none of us have training in mental telepathy, as someone noted. In an Emergency, tell us what you need, and we'll do our best to see you get it.
