Both the above are right.
There's a "two out of three" rule. The three are:
- country of reg of aircraft
- country of pilot's licence/certificate
- country being flown in
As long as two are the same, you're OK.
So you can fly an F-reg or a D-reg in its home country on your US PPL, but you can't legally take them abroad.
To cross borders, you need an N-reg.