Good advice. Don't rely on anything from your company. Ask Oasis/XL/Futura pilots about that! Captains do not mind signing- remember they had to do it themselves! The Captains name ALWAYS goes in Commander box. It you are PF, I think you may log as P1u/s though some say you should not. I think if it is your leg and you take the decisions, then P1 time applies, though understand some disagree with that and say you should always log P2 unless you are THE Captain. I have heard that really only one person should log P1 for any flight.
I think it better to log each leg and time. Reason being that it is not unusual to be phoned with flight recorder queries or rockets, and exact information about times and places may get you off the hook.
In short, it seems there is not much hard and fast guidance. Common sense is enough.