BEagle,
There are a number of definite references quoted above to support the view that P1/S is ONLY to be used for flight tests with an examiner, yet you are countering by merely saying that this is not true because it's "obvious" or a "basic concept".
It does seem that this 'basic concept' - that you can log any supervised flight as P1/S - is basically WRONG. The references quoted are pretty clear and unambiguous - P1/S is for flight tests, PICUS is for two pilot aircraft, the only other possibilities for normal flights are P1 and PU/T. The rules are there to stop people wrongly claiming P1 time;; you do seem to be suggesting flaunting these rules!
Please do correct me if you have a reference that says otherwise, but I'm afraid I just cannot trust a bland statement of "it's obvious".
As for the matter of whether someone undergoing a club rental check should log P1 or PU/T.. I too would love a definitive reference on that! though the answer from the CAA bod above does seem pretty sensible.