The case headers you have are called ‘keyed’, “shrouded” or ‘boxed’ headers, and they force the cable to connect in one orientation, as you mentioned.
I think your cable is offset by at least one pin, going off of the image you posted. Instead, the red stripe (-12v connection) should be all the way on the bottom, with three rows of pins on the other side left empty:
Often when a module uses only 10 pins of the connector, a 10-pin to 16-pin cable is used, with only 10 strands of cable (as is the case for the module connected next to the nRings module in your image).
The nRings says ‘either way’ because it detects which line is +12 and which line is -12 and won’t be damaged by reversed power. But for example if you connect the power connector one row off to the side, you could accidentally connect +12 or -12 to ground, which can damage other modules in the case.