When I turn the dimmer to the off position (max resistance), the bulb
turns off completely. Does the lamp still use power in this position?


yes, and quite a bit too. Dimmers are often described as devices for
turning down the efficiency of light bulbs. No light output comes from
somewhere roughly in the region of 50% power consumption.



Unless the triac is not triggering at all. I'll have to experiment with
some dimmers and see what happens. Easy enough to measure and find out.