There is no "trim in motion" sound on the 737, merely the generally noticeable sound of the trim wheel spinning. The "sound that explains everything" is the 5 seconds of the trim wheel running. When I am next working I will have a look how much of the trim range 5 seconds of trim at high rate covers, but I would guess it is not far off half the total trim range.
Most who fly the 737 will subconsciously be aware of any trim movement, and look to see what caused it, especially after the Turkish accident, which I suspect almost every 737 driver has relived in the sim. In that case, the biggest clue (apart from the speed and the speed trend dropping through the floor) was the trim wheel running considerably longer than it would in any normal condition.