If the instructor is logging P1 then you must log DUAL or P/UT.
If I fly with people as an instructor I am considered to be in anyway in command then I log P1 and they must log DUAL - that means ppl/cpl/atpl on a club check or a student on a course such as IMC, NIGHT etc...
PIC U/S or P1S is ONLY available as the result of a SUCCESSFUL flight test with an examiner unless PREVIOUSLY agreed with the CAA.
As no-one I have ever known has this agreement any flight where the instructor is logging P1 means you must log DUAL.
The CAA guidelines are VERY clear on this. Used to be in CAP53, GID44 and now LASORS. It has ALWAYS been the same.
Excuse the tone but it really annoys me that I get PPLs argue with me about this - and worse still a LOT of instructors get this wrong too...the documentation is quite clear....
Anyway..hope this helps clear it up.