P1/S, P1U/S, PICUS, call it what you will, is for the result of a successful TEST
If you refer to LASORS then yes, that is correct insofar as you are talking about a flight for the purposes of license or rating issue. LASORS does not recognise training that is not for license/rating issue and I really don't know what the legal situation is as I suspect that are not PU/T if you haven't been receiving recognised training.
So for something like a club checkout, either the instructor logs the flight as P1 and you as SNY or you as P1 and him SNY or go for PICUS. I haven't found anything written anywhere and the CAA seems to accept the odd hours of such flights as PICUS (in my case at least).
P1/S when conducting differences training in other aircraft after I got my licence.
Definitely not P1/S as the case is covered in LASORS as
Pilot under instruction for the purpose of gaining a licence or rating, or for conversion to an aircraft type within an aircraft rating group or class.