The instructions as to who records what as P1 time on type on augmented or heavy crewed flights varies a little from jurisdiction to jurisdiction. What all jurisdictions agree on is that the PIC may record the entire duration of the flight regardless of what seat they do or do not occupy. Where the various disagreements often occur is what does a Captain log who is not the PIC. There is only one PIC. Most jurisdictions agree that a Capt, who is not the PIC, may log P1 when occupying a control seat in the absence of the PIC. Even here the boundaries may get blurred a little at times.
Keeping a personal log book is a little like keeping your score card at golf. If you play a round on your own, and have 84 off the stick but record that on your score card as a 72 - then you do no one any harm per se, as you are only fooling yourself. If you later pull that card out in the clubhouse and tell everyone you had a 72, and have the score card to prove it ......... now that's a different matter.
