Your description of events tells me that the Arduino wiring is not directly part of the problem. You have confirmed the wiring change is working by voltmeter testing and by Arduino logic testing. Which implies that there is another cause of the motor not running under throttle control.
1. I checked that the "GND" is common for everything (everything on the bike, the 12V DC-DC converter, the arduino
2. The input of the DC-DC converter that makes 12V is connected to the lights switch so I can turn it on/off easily
I think your DC-DC converter is putting too much load on the tiny common ground and battery feed wires in the main harness. These thin wires will work like a resistor and will have a voltage drop along their entire length which will vary according to the current they are carrying. The voltage drop will be equal to the current flowing through the wire, multiplied by the resistance of wire.
If you measure the voltage at the battery and then measure the voltage at the inverter connections
(while it's switched on) you will probably see a noticeable difference in the meter readings.
If you have 54V at the battery, but only 50V at the inverter connections, it means there will be ~2V drop on the thin battery feed wire in the harness
and ~ 2V rise above 0V on the common ground wire in the harness.
I had a similar problem when I wired a 3w LED light
(with a built in inverter accepting 12-80V input voltage) directly to the lighting feed and ground wires of my MP4 wiring harness.
In my case the current draw from the LED was only 85mA, but when the light was switched on, it was enough to cause a significant voltage drop on the battery feed to the LED battery gauge
(causing The three LEDs to dim slightly) as well as a corresponding voltage rise on the ground wire which effectively caused the motor to run without the throttle being touched.
The voltage drop on the feed wire could affect the gauge operation, but it should not have affected the throttle operation.
But the increase in voltage along the common ground wire meant that the throttle hall sensor had +5V on one side and probably +1~2V on the ground wire instead of 0V. Therefore the signal output voltage could also be 1~2V higher when measured in relation to the ground connection on the controller.
On the MPIII, if the controller is switched on while the throttle is operated, there is a safety feature which prevents the motor from running until the throttle signal voltage returns below~1V. I suspect your DC-DC converter is causing an increase in the throttle signal voltage, which could activate this safety feature.
Disconnect your DC-DC converter from the lighting wires and connect it directly to the battery wires
(to avoid overloading the thin wires in the main harness) and I think you will find that the throttle will then work normally.
Alan