FAA - any flight with an instructor can be logged PIC (if you're rated in the airplane)
Being a US/FAA flight instructor myself, I don't agree with "any flight with an instructor can be logged as PIC". Basically one must hold at least a FAA PPL and must be current in the appropriate class and category of aircraft in order to log PIC time. During PPL training the student pilot can log solo time as PIC time. After obtaining the PPL the pilot can log PIC time even with a flight instructor aboard as long as the pilot is current and the flight takes place in the appropriate class and category for which the pilot is certified.
FAR Part §61.51 Pilot Logbook (e) Logging PIC time
https://www.ecfr.gov/cgi-bin/text-id...1_151&rgn=div8