One quite amazing aspect of this thread is the number of (apparently professional) respondents arguing on the basis of flag waving and 'my country's ops are better than yours' nonsense.
There is no need for that. This case is very straightforward.
In an emergency, declare: "MAYDAY, MAYDAY, MAYDAY ... [FULL CALLSIGN]" and follow up with details of the emergency.
If the incident is less serious, declare: "PAN, PAN, PAN ... [FULL CALLSIGN]" and follow up with details of the emergency.
This will ALWAYS command the full attention of the ATCO, I assure you. Regardless of national niceties. Just do it!
Anything else risks confusion as this case illustrates. Remember that this was an UNEXPECTED non-standard call for the ATC staff to interpret out of the blue, so it is vital that it is as clear as possible and makes sense. At many ATC units, staff will flick the RT onto loudspeaker on hearing the "Mayday, Mayday, Mayday" prefix to a transmission so all present can hear the message and get straight to work helping out.
Use those three vital words, please! Save the colloquial chatty RT for routine situations.
By the way, I have always considered aircrew from the US and Europe to be very competent and professional across the board. Well done. But don't defend mistakes ... learn from them! (ATC and aircrew alike).
SHED.