This is not the way to do it.
You need a splitter cable ("Y") cable so that you can plug the 12Vdc into the External Power jack on the UNO.
Two separate cables would also work but the +12V for the UNO should be plugged into the External dc jack.
I seem to be able to run both the UNO and motors using a 12 volt source plugged into the motor board.
Without a photograph I don't know how this could be working unless you are using a SHIELD and failed to specify that. "A motor board" does not mean "SHIELD" to anyone because if it was a SHIELD, it would have been called a SHIELD, and not a motor board, therefore we understand you to mean a NON-SHIELD motor driver. If this is not the case you need to be more specific. If you meant a shield then the +12V jack on the shield is probably connected to the Vin on the UNO and it is the same as a separate cable without the cable. Please specify or post a photo.