It would be a good idea to get an oscilloscope on line L33. I think that you will find that the brightness is controlled by PWM, not by a change peak voltage.
I guess that the 9 V is what you measured with a voltmeter. That would imply that the power is on for 75% of the time. However I suspect that the power is on for a lot less than that, because 75% would appear nearly as bright at 100%. I think that during the parts of the cycle where the power is off, the voltage doesn't fall all the way to 0 V, so the average voltage is much higher than the peak voltage multiplied by the duty cycle.
Vehicle circuits have to work with a range of supply voltages, and the change from 9 to 12 V would be badly affected by the battery voltage changing.