Thanks Bose, for clearing that up.
I do disagree though that your comparison of a European pilot working in the US and converting doesn't quite stack up.
We are talking about two completely different things.
I am a European Pilot / Citizen who chooses to fly 'N' reg aircraft which happen to be in Europe. I have made a living like this for many years. I did not require a European licence to fly before and the FAR's do not state that a European licence is required to fly 'N' reg aircraft.
It's like the FAA specifying that you need an FAA licence to fly a European aircraft on say the German D register, that happens to be based in the US (even though you have a European licence) This of course wouldn't happen.