You are probably correct about the voltage spike, and for just that reason, we strongly recommend to use separate power supplies for motors/servos/solenoids and Arduino. Don't forget to connect the grounds.
If separate power supplies are unavoidable, you need excellent power supply decoupling for the Arduino portion of the project.
Depending on the direction of the motor rotation, you might have a 12 volt spike that adds to the 12 volt supply and likely killed just the voltage regulator on the Arduino board. When powered up, check the input and output voltage of the regulator chip.
Another option may be to use a buck converter the convert the 12 volts to 5 volts and send that to the Arduino 5 volt pin.
Paul