Agree with all of the above Genghis but with the engine running you are excercising the priviliges of your certificate and acting as PIC, captain of the ship, the big kahouna, whatever the title is.
You carry the responsibility you can log the time.
Now I have to admit this was all much more important when my time was still in double digits then it is now.
Part of my duties are now to play the logbook-police and make sure these little critters don't log stuff they're not entitled too.
You log what you pay or you get paid for what you log, either way.