Years ago, I had a "situation" in a light aircraft that had taken off from BOS on a VFR flight. I told TRACON that I needed an immediate return to the airport (we were about 1500 feet, 2-3 miles from the airport).
"Are you declaring an emergency?" the busy departure controller asked.
I hit the ball back over to his side of the net. "Not if you'll clear right back into the field," I replied.
There are times when the communications about an emergency need spelling out more clearly, as was the case with the unfortunate Avianca 707 fuel exhaustion crash on Long Island as they accepted endless vectors on an approach to JFK, eventually running out of fuel.
Sounds like the communication was superb here with the US Air birdstrike even without the Mayday call.