The 90 day rule simply states that you must perform 3 take off and landings as sole manipulator of the controls in order to carry passengers.
Makes no difference if you were P1, Pu/t or P1/s as long as you were sole manipulator.
In terms of logging time with an instructor when you have a licence and are just doing refresher training or club checkouts, you really need to clarify with the instructor before hand. If he's happy to just be there in an advisory role, you can sign out as PIC and log P1. If he puts his name as captain when booking out, and he logs the time, you can only log Pu/t (pilot under training).
P1/s or PICUS (pilot in command under supervision) is a special case which only applies to a successful skills test. It is often incorrectly used for a club checkout with an instructor, or the typical circuit check required if you haven't flown for 28 days and want to hire an aircraft.