Pilot Flying is a particular piece of terminology associated with two crew operations so I wouldn't use it in this context.
On a private flight if the commander is happy for someone with no status to handle the controls, then that's his business, as long as he recognises that he is the commander and ultimately responsible for the flight.
On your last point, most UK AOCs define positioning flights as CAT, and within PART-OPS there is a prohibition on unqualified persons handling the controls.