Some random thoughts on logbooks in no particular order.
1. It's your logbook, not a flight schools, not a CFIs. As such, you are responsible for correctly logging time. It is a legal document and YOU are held liable for errors and falcifications, not your CFI. As a CFI I may help the student pilot figure out how to log the time; beyond that it is the pilot's responsibility. I may show/explain what the FARs say, but don't correct his logbook. I'm really fed up with CFIs crossing out PIC time logged by private pilots in tailwheel, complex and high performance airplanes and instrument training in which I give them instruction.
2. Different entities may have different ways of tracking time that are not IAW part 61. If so, that's fine- yet you must still log the flight time in your logbook IAW part 61. For example, the military counts certain time that the FAA does not recognize, flight time does not start until wheels up, and some flight time can not be double logged. For example, as an IP (instructor pilot) in the military I logged instructor time while evaluating from the jump seat. I had to log this time in order for there to be flight records that jibbed with training records. If there was training recorded, there needed to be a flight log to back it up. Even though the military tracked this time, I never logged it in my FAA logbook. I could log only night or IMC in the military, not both. My FAA logbook shows both.
A school may have a very good reason for not logging this time as PIC in their records. What the school tracks has no impact on what you may log. If your time is legal as PIC, log it as such in your logbook.
3. The CFI is signing off on the training performed, not the type of flight time. Log it as PIC and if the CFI refuses to sign the logbook, talk to the FSDO. CFIs can and have had certification action taken against them for refusing to sign a logbook. I'll try to find the case tonight.