What your instructor was probably trying to get at is called PDPIC time. Performing the duties of pilot in command. THIS IS NOT PIC TIME! It is simply an endorsement to satisfy the FAA requirements of "Solo" cross country or "solo" time in a multi. However, your instructor is always going to be with you. I usually make an column labeled PDPIC or have an asterisk next to every PIC time logged.
On the 8710 you log it as PIC and solo time, however your log book should not match this. Because in the FAA's eyes this is technically "solo" time.
You can not log PIC in the aircraft unless you are rated or sole occupant. Which in your case, is neither of those.