Airlines (part 135 and 121) only have one PIC and one SIC. The Captain logs PIC and the first officer logs SIC. It doesn't matter who's typed and who's flying. I'm not sure about relief officers on long international flights. They are TIC (third in command), so I guess they need a new column in the logbook?
I don't need to "log time" anymore, but I do it for fun. I'd like to see many hours I can get when I retire. I have 11,000+ and 6 airline type ratings with over 20 years to go until retirement, so I could get another 15,000 easy. My company keeps records too, so I can get my flights off my computer at home and log them. I keep a small log (like the popular little red book) and get my times off the ACARS to log. I check the company to see if they change the times later. Yes, they have been changed before.