Don't complicate this!
Just put in your logbook whatever your operator is putting down in their log. Remember that SNY hours generally doesn't count towards your annual/monthly/weekly accumulated FDP.
If your operator logs one thing and you log something different, you will end up with a discrepancy. You really need to make sure you have the same records as your company as there's a mutual responsibility for keeping legal on FTL's.
The co-pilot/first officer is licensed to fly the aircraft - licence, medical, type rating (including base training) all valid/completed.
It does not matter if you're
licensed to fly it. You are still not
qualified to fill whatever position (PIC/SIC) on that particular flight unless you've been signed off for it by your operator.
A direct entry captain is unarguably both licensed and competent, but he can still not act as PIC until he's been line trained and line checked by said operator.
A F/O (even a SFO) can still not be the SIC unless hes been signed off to fly without the safety pilot. Yes, he's the one manipulating the controls, but hes not the SIC. That privilege is filled by the safety pilot.
Also remember that whatever your company is doing, that's what's agreed/approved by the authority. If you depart from this, no matter how much sense it may makes to do it differently, you are effectivly in violence of the procedure.
If the procedure is for a pilot to be officially signed off by operator for something he's already licensed to do, then that has to be done that way. If procedure is to log flight hours in a particualar way, then it has to be done that way.