Does the switch simply push into the brake lever holder on the DT360a, or does it turn when in?
If so, what actually holds it in place as mine seems to be pushed out by the lever when operated.

There should be one or two small tangs on the switch body. There will be at least one corresponding hole on the brake perch with which the tangs line up, and the tangs depress on entry, then click out once fully in - this is what holds the switch in position. To remove, you need to depress the tangs and pull the switch out.

If the tangs are just flattened, gently pry up with a small flat blade. If they are missing (ie snapped off), then you'll need a new switch.

I recently put one on my 125 and 360. Don't recall a twist being required. Recommend clean up the perch well before install.