A guess here but take a look at the fan switch and see if you have 2 wires connected to the "motor side" of the switch or an extra wire at the motor.
What I'm thinking is the problem light is getting it's ground through the fan motor. That could explain the motor running at random as it is also getting 6V through the problem light when that switch is "on"
A test would be to turn on the problem light and check the fan to see if it running at half speed or even slower.
The logical explanation is the bulb is getting it's ground from an unknown source. Having said that many times electrical circuits do not behave in a logical manner.
Whatever the problem let us know when you get it worked out.