Whenever I start the engine, I tap ‘Log Engine Start’. When I start to taxi (from hangar to apron) SD automatically starts counting the off block time. When I stop on the apron and let the engine warm up, the off block time counter stops counting, restarting to count when I taxi to the holding point.

SkyDemon should log "off blocks" automatically when you first start moving after logging the engine start. It should not log "on blocks" until you stop for the final time after your flight.

My observation is that the block time-counter stops counting while you are not moving on the ground and resumes when you start moving again. The counter however seems to continue running in the background even while you are stopped. So in the end it displays the correct block time as defined by FCL.

That's right. It stops counting because if you remain stopped, that was the correct time. As soon as you start, it knows that wasn't your final stop so keeps going from where it left off.