Problems while using accelerometer as a motion switch
The device containing the accelerometer is experiencing a force (of x G) momentarily, against the direction of acceleration, when the motion starts. Then it will experience 0 Gees as acceleration stops, i.e. once the target speed has been achieved. 

Similarly, when bringing the device to a standstill from constant velocity, it will experience a momentary force opposing the deceleration, then zero force again once it comes to a standstill.

That is what your experiment reports - the LEDs flash when the acceleration is experienced, once in each direction. In other words, the behavior is as designed. 

For keeping an indication on as long as motion continues, an accelerometer is not the ideal device. It is not a motion detector, it is an acceleration detector, as the name implies. 

While some simple software trickery (integrating the acceleration over time) could be used to track when acceleration is felt in one direction or the other, such a mechanism would fail if the deceleration (or acceleration) force is very low compared to the sensitivity and sense signal noise floor, such as if one were to accelerate the device very sloooooowly, but bring it to a halt rapidly, or vice versa.