I would proceed a little differently. Since the spacing between the sensor and the magnet on the cam gear is critical, I'd unplug the connector and insert some very fine wires into each pin position (just individual strands from a scrap piece of stranded wire), then refasten the connector. Don't force it if it won't go back together. Keep strands separated so there are no shorts. Then use your voltmeter and try to detect the signal pulses each time the magnet goes by. I'm not sure, from the diagram, which two pins you'll find the signal on and since it is a narrow pulse, the meter will probably just blip. Looks like I'd connect the ground of the meter to Pin 1 and the positive lead to Pin 2 for starters. (Make sue Pin 3 is carrying +12 volts with respect to Pin 1.) This would at least give you an indication that the sensor was or was not functioning. Good luck.

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You will not see an AC voltage signal on this type of sensor. It is a Hall Effect sensor and it triggers DC On/OFF. That is the advantage of this 3 wire sensor over the Induction , 2 wire sensors [ they are AC V output]

Your sensor has a 12 V feed from the ECU [+ and - to power the IC of the sensor] and the third wire is the triggered on/off signal back to the ECU.
So, you want to see a constant 12v at the + and - pins, and a pulsed, DC signal at the third pin, which is the sensors switched output signal to the ECU. This is a switching sensor, unlike the 2 wire AC voltage generator sensor .......

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A Dalton