Hi Augustin and
to the forum.
I experienced a similar problem on one of my bikes when I installed a 10 Watt LED headlight, but that was with due to the thin wires used in the wiring harness.
The 48V LED light and the 5V throttle both used the same ground wire, and the additional load when the headlamp was turned on would cause the motor to run.
When I measured the voltages on the throttle wires I was surprised to see a voltage of 0.84V on the throttle ground connection when measured relative to the 0V battery ground connection.
This 0.84V increase on the throttle ground wire caused the throttle signal voltage to also rise enough to cause the motor to cut in.
More details can be found
here.
If you are able to measure the throttle signal voltage at the controller's connector, see if it increases from 0.8~1.0V to more than 1.2V when the 12V circuit is live.
If you have a spare unused cable in the multi-conductor cable, try running a separate ground connection for the throttle to see if it will cure the problem.
If the motor runs, but the throttle signal voltage remains below 1.2V, it might be an electromagnetic interference problem if a DC:DC converter
(or LED Driver) has been fitted
(assuming you have not changed the default 1.2V value for the Throttle minimum valid voltage (V) setting).
Check out
this topic for more details.
Alan