IHD, read the answer above mine. The important bit is "...the a/c moves under its own or external power.... As a pilot you log block time unless you are an air force type who only logs airborne time.
It may be silly to you for someone to log the time whilst they are still connected to a tug but them's the rules. Last night I landed at JFK and thanks to the complete gridlock that the ramp controller managed to make, it took over an hour to get to our gate. That's an extra hour for the log book and had I been a keen young new-hire, that'd be very nice thank you for building up my totals.
Block time is from 'off-blocks', as long as it is for the purpose of starting a flight. Whether pushed back or just taxied straight off. 'On-blocks' is when you finally come to a stop and set the parking brake for the last time before you get off. At some airports, we have to stop before we get onto stand, shut down the engines and wait for a tug to tow us the last few metres onto the stand. Until we are on the stand and the brakes are set, the block time isn't over.
As for flight time, that's fairly obvious. Our on-board systems record both block time and flight time and give a very nice little print out at the end of every sector!