If a pilot is within the currency rules as laid down, then they can fly with another PPL as a safety pilot, but only they can log the time.
But, If any instructor is with them, then they are PIC and the PPL must log P/UT. There can only be one commander on a flight.
If I am on board and responsible for successful flight, then I log the time. If any training occurs, then I am instructing, therefore I am PIC.
A checkout is usually because someone hasn't flown very often and invariably turns into an instructional flight either on my or more often on the PPL's urging.
I never understand why people get in such a tizzy about what to log. It's not as if it makes much difference to the recreational flyer. Your total time goes up the same whether you log P1 or P/UT. So if someone has done well, then I have no problem in not logging the time, but if it's a bit ropey, then I will say so and try and help them fix the problem. It is my job after all.
A check flight isn't the same as a solo flight, mainly because the the person handling the controls, isn't in charge of the flight in total. The FI is directing the PPL.
That is a totally different thing to just having a safety pilot on board.
Currency is the most important thing for flight safety and club rules are usually as they are because of bitter experience. Most PPL's are very sensible and understand when they are rusty, but there a few muppets out there who seem to think that flying with an FI is bashing their ego.
Use your common sense and if you do have to fly with an FI, then get the most out of it, after all you are paying for it! Get them to show you something new or re-explain something. It's far better than just having us sitting there bored.