How do these two sensors work?

Question

Electronics

Why is it fed like that?
I assume that the sensor works on the principle of potential difference or what?
And how will a passing tooth cause this current to start flowing?

And how does the metal passing by the sensor (not a magnet) open the transistor and where is the 12 volt power connected here?
Here, the control unit fixes the signal at the moment when the current begins to flow from the control unit to the ground?

And which one is used for what purposes (or where) or is there no difference in them?

Answer 1

if it is clumsy, then the sensor takes power from + and -
the third output, when the magnet is brought up, is shorted to the ground, regardless of external connections. in the circuit, a high potential is applied to this output, respectively, when triggered, the potential drops to conditional zero, which is fixed by the controller.
a hall sensor or a "metal presence" sensor is the same for the user, but it is better to use a hall sensor.