NS - if you search through the archives you'll see that this topic comes-up regularly.
Deciding how to log a flight is simple if you ask yourself a few questions: who signed for the aircraft, who will be interviewed by the CAA if you bust airspace or break a rule, and who will fill-in the paperwork if the aircraft ends-up in a hedge. Whoever it is, that person is the Captain and logs the flight accordingly.
If I am in a school aircraft as an instructor then I am the Captain. If I am flying with a private owner who wants to polish some skill then we will agree beforehand who is the Captain and who will deal with emergencies.
Regarding the P1/S suggestion - this might appear to be a logical answer but the rules are clear: in SPA it can only be used for a successful flight test with an examiner. It's unfortunate that the CAA have not relaxed the P1/S definition to include checkouts with an FI as this would remove a grey area..
HFD