My log is mostly for me so I remember sites I've been to while traveling. I only put very basic dive info and anything really noteworthy about a dive, if there was. For a trip I also jot down general things like the dive op, cabin or room numbers (where we stayed or maybe the nicer looking one I want to ask for next time) or even restaurants that I might like to remember when returning to a place. Sometimes a note when I changed the computers in my battery or other maintenance things, as I'm always trying to remember that. I find that non-standard stuff more useful than the usual. Early on I recorded stuff like weight, when I was still sorting things like that out and tended to forget in between so that was helpful.
I've never had anyone ask for a log, as mentioned people could make these up anyway, but I'm sure someone out there does. Also haven't had anyone sign it in ages, only the first couple dozen dives, or dives part of a course. But it's really quick to log the basics so you have something, even if it's just noting the location with time and max depth from your computer. May as well. Beyond that I'd log what's useful for you, not get writers cramp filling out all the slots in a standard logbook just because they're there. I made my own log as any I saw had lots of space for stuff I didn't care about and none for stuff I did. Many people do that and have posted samples on this board.