Am I using a multimeter incorrectly? Or is there a short in the cable?
You are doing correctly. All grounds should be connected together and should all read the same. The readings you are getting indicate that the ground in your cable is disconnected (“floating”). You are getting non-zero reading due to phantom voltage. That is to say, your volt meter is high-impedance and so picks up stay energy.
To verify this, you can do the following with the power turned off (I’d suggest turning off the main breaker): switch your meter to resistance and connect across ground and neutral in your cable. These should be tied together back in your main panel and should read close to 0 ohms. I suspect you will see a high resistance or open (display just shows “1”).