A logbook is your own personal property so you can write what you like in it - as long as you don't use the total # to obtain some advantage.
I know a man who logs passenger flights with me, and it's not my business what he writes in there. Also lots of people "log" their visits to the CAA safety seminars; they lay out their logbooks open on the table so the CAA man can apply his official gold plated rubber stamp to the logbook, and they can go home with a nice warm feeling that they are now 23.5 times less likely to have an accident
Incidentally there ARE instructors that have only a PPL. Up to about 20-30 years ago this was allowed, and from what I've been told one could get grandfathered rights to what became a BCPL, then one could even get an IMC Rating granted on the basis of that.