If the engine is running or not should not impact the voltage drop across the resistor. The voltage at the coil side of the resistor should always be less than the feed side.It sounds to me like the alternator is somehow feeding full voltage to the coil through another connection.
Start the car with the alternator disconnected and see if you still have a voltage difference across the resistor. If you do you need to track down how the alternator is back feeding.
Could be as simple as an incorrectly wired ignition switch.
An easy test would be to disconnect the everything from the + side of the coil except the wire from the resistor and run a temporary feed from the battery to the other side of the resistor and start the car. See if the voltages are normal (14 on feed side and ~9 on coil side)at that point. If this looks normal with the car running look at the second wire you had disconnected with the car running and see if there is voltage on it. There should not be. It should only have voltage while cranking. If there is voltage there you need to find out why. Could be as simple as the connection on the starter, which is where this should go is somehow shorted to the battery lug.
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