This thread seems to be following the finest tradition of PPRuNe inasmuch as a flat statement is followed by a flat contradiction which is then followed by personal abuse.
To my mind, the essential points are that only one person can log P1 in a single-pilot aircraft and the commander has to be designated before the flight - in a club/school environment it is he/she who signs for the aircraft.
The latter point is an absolute essential (and I'm speaking with my ex-CAA hat on now) since, in the event of a fatal crash, this is where the investigators will decide who was the commander. I can't speak for the ways of the FAA, they do things differently over there (not better, or worse, just differently) but in the UK there is no capacity to switch commander mid-flight. Thus, in the case at issue, the checking pilot has to be in command because he may have to override the pilot being checked.