Logging of 'Flight time'
suggest you look at Lasors, Section A, Appendix B. Where it is specifically noted how your flights should be logged. If a flying club subject to its rules require - simply that an Instructor accompanies you to observe whether they consider you competant to hire their aeroplane, then you are the PIC simply being observed and log it as PIC. Should the flying Club require that all hirers undertake training on an aircraft prior to hire then it must be PUT.
As to the Biannual Flight Review your local FAA office will answer your question as will the FAR on how such flights are logged.
Additionally a succesful flight for the Issue of or a Renewal/Revalidation of a license or Rating (a Test) should be annotated as PIC/us and entered into the P1 column. The Examiners name in the name column. An Examiner should log the flight in their own log book as PIC and in the P1 column also.