I have two experiences that may be worth adding here. In the late 1980s, I carried out a three-engine ferry flight in a Boeing 747 from Trinidad to JFK. It was planned as such, together with the appropriate remarks on the ATC flight plan. On nearing New York, the controllers asked whether I was declaring an emergency. I replied, no, because it was a normal three-engine ferry being done in accordance with company procedures and UK CAA rules. Nevertheless, they insisted on declaring an emergency for me and so, after a normal landing, when we left the runway, we were surrounded by emergency vehicles.
Some months later, I received a rather peremptory letter from the FAA asking why I had entered US airspace with an engine out, why had I not declared an emergency, and saying that I had broken FAA rules. I wrote a polite letter back saying we had filed the flight plan correctly and it had been accepted, there was no emergency, everything had been done in accordance with UK CAA and company rules - and aplogised for any inconvenience we may have caused. I never heard anything more.
On the other hand, many years before, in 1966, I had been involved as the co-pilot in a rather nasty experience on the climb out from JFK. We were at around 20,000ft, when the aircraft, within seconds, rapidly filled with dense smoke which reduced the internal visibility to only 6 inches. At the time, we thought we were on fire and the captain started an emergency descent during which, for a short time, we lost control and all radio comms - we were convinced we had a major fire on board. When were able to re-establish R/T communications, the controller asked the nature of our problem. My first words on the R/T were totally non-standard, "We are full of smoke and can't see a bloody thing."
Bless New York ATC! They reacted perfectly, having cleared all other aircraft off the frequency and then gave us vectors back to JFK that were neither too fast nor too slow for an over-weight landing. After landing, we were surrounded by emergency vehicles and, as by then there were no signs of fire, they escorted us to the ramp.
It turned out that one of the cabin airconditioning compressors had lost a seal, emptying all its oil into the hot air stream where it was instantly vapourised into thick white smoke.
Despite my non-standard R/T in the stress of the moment, I was so very grateful to those controllers!