Do you have aftermarket levers?I took another look at it today. When I squeeze the lever, it initiates the rear brake light. As I pull it back and stop it still stays lit. But, if I let of of it just a little bit, it turns off and does not turn back on if I squeeze again. All of this is within the same pull.
I had the same problem with my levers. There is a little electric switch that contacts with brake lever. When you pull the lever in, there is a spring loaded plunger on the switch that pushes out. When it comes out, the switch turns ON and the brake light illuminates. When you release the lever, the lever pushes the plunger back in and the switch goes OFF.
There is likely a bad interaction happening with that switch when you pull the lever out. That's what happened with my levers. I had to grind a bit of the lever away where it meets the plunger to make it work properly.
Also check the operation of that switch. If it's not working properly, replacing it should fix your problem.