yes, there is a little magnet inside the control that triggers the brake light. somebody else with more experience with it will tap in shortly.
Yes Correct. This switch can move if the small set screw is loose. This can cause the switch to not function properly giving you no brake light function at hand lever. However, foot pedal will still give you brake lights.
Another cause could be a bad switch or bad wire.
Pull the switch housing apart. You will find the small Allen set screw at the bottom just below the magnet switch. Loosen the set screw up and move the switch back and forth while watching your tail lights to see if the brake lights come on.
If you get the lights working properly, you want to set the switch before putting the housing back together. To do so, move the switch until the brake lights turn on, then back the switch back out till the lights turn back off. Move your lever to verify the brake lights are functioning correctly. But, don't have it so close that the brake lights turn on with very little lever movement. If set too close, they can possibly flash on and off from vibration while going down the road.
Once you get the switch set. Tighten set screw back up and install switch housing.
NOTE*** Do NOT over tighten set screw. It will damage the brake light switch.