Is your logbook for the FAA — use the FAA's rules to meet the FAA's requirements for certificates, ratings, privileges and currency as 61.51 says — and then report the data based on your potential employer's standards (or create a separate colum for what you think they are)?
Or is your logbook for a company you may or may not have an interview with in the future — use what you think are the company's experience requirements to fill out your logbook and maybe miss a qualification for something along the way or have a possible falsification claim because you "signed for the aircraft" but weren't authorized by the FAA to log the time as PIC?