Yes, those two wires (the white painted black and the black you point to that I labeled "switch") are in the same romex line.
I am using a VAC/DC 110/220 tester that has a black and a red probe with a light that turns on inside when it responds to current. It's a little hand held item a little bigger than a pen.
When I used the tester:
switch off: white + ground = tester lights up
black + ground = no light on tester
switch on: white + ground = light
black + ground = light