The cause is the disc brake pad spreading spring contacting the brake rotor. Apparently the slight squeeze of the brake lever changes the geometry of the mechanism so that the spring is contacting the rotor continuously rather than intermittently, and thus the click goes away. When braking hard, the spring is squeezing so hard on the rotor (SM-RT64) that the uneven circumference of the rotor again starts to click as the spring touches the edges of the rotor intermittently (the rotor is not a perfect circle but slightly deviates from a circle, so the "hooks" of the rotor edge can touch the spring).
I changed the brake pads to new ones and the sound went away.
This click, by the way, seems to be a very useful "pad wear warning". If the pad spring start to click before pads wearing to the backing plate, the user knows to replace the pads before they are dangerously worn.
When inspecting the old brake pads, the initial diagnosis about some pad life left still seems to have been based on one location of one pad. The left pad had been worn far more than the right pad, and the pad wear seems to be uneven on the left pad, so that one location of the led pad had more material than another location. So quick inspection without removing the pads didn't reveal the full situation: that one location of the left brake pad only had 0.5mm of material in it.
And a final rant: disc brakes seem to have poor pad life even in perfect conditions (the bike has practically never been ridden in the rain). I have never experienced as bad pad life as 2000 km with rim brakes. Yet, with disc brakes short pad life has become the norm. I may have to test sintered metallic pads to see if the increased pad longevity does not cause a major rotor wear problem.