Hi,
sorry your description is some how confusing.
I assume you are using your proximity sensor as a low side switch (minus terminal is connected to ground).
You are talking about a LED, so I assume you have connected a LED and a resistor in series to the plus terminal of your sensor, and the +12 V to the other end of this series connection. Or is there a LED bulit in, as indicated by a small note in the construction drawing saying "action display light (RED)". Further, the picture of the sensor I have found shows 2 wires, whereas the datasheet talks about 3 wires.
According to the datasheet your sensor is normally closed (NC), so if there is no mateal in its vicinity the LED should light and stops to light
Please show us a sketch of your circuitry, indicate where you are performing the measurements. Based on your description I'm not able to give any advice how to solve your problem, or clearly identify your problem.
BR