Just to clarify the situation, this device is just an input sensor. When you press the lever, it gives out a voltage proportional to how far you press it. The voltage is 1.2v at rest and 3.8v fully pressed. The motor controller converts that voltage into a number. What it does with that number depends on the software in the controller. The programmer could do anything with it from flashing the lights to working the brakes. In other words, it isn't connected to the motor, so has no direct influence on it
The three common ways that it's used are as below, but I've only ever seen one at a time, which is a shame because 1 and 3 combined would be very useful for those that can't pedal hard, and it would be completely legal. Are you listening Wisper, Woosh, Juicy, et al?
1. Start assist. when you press the lever, the motor starts and goes up to 6km/h, then holds that speed.
2. Full range proportional speed control, where the motor will try to spin to a speed proportional to the distance pressed from zero mph to the motor's maximum speed.
3. Pedal assist boost, where pressing the lever overrides the pedal assist power setting to give either maximum assist or proportional speed as in 2 above.