Certain hall switch operation

You're probably suffering from bounce. If you were to connect an oscilloscope to the Vout pin (I suggest you try this) you might well see something like this:

When you plug or unplug the jumper, it will be almost impossible to do this without causing several unintended pulses. I recommend not trying to program it by using jumper wires.

By far the best way to do it is to use a microcontroller and some transistors. However, if you really want to do it by hand, using this circuit might help:

Here, you can press S1 to pulse Vout to 27v, and S2 to pulse it to 3v. C1 will help to remove any switch bounce. I don't guarantee that this will work though.

The HAL508 and Optek OH180U are both Hall Effect SWITCHES with hysteresis. The HAL has an open drain output ant the Optek has an open collector output, both on pin 3. When you look at the magnetic specs in the data sheets, after converting Gauss to milliTeslas, you'll see that the two sensors are very compatible.

Since switches are on/off, I assume that the function is to say whether the Hi Hat is open or closed, not to read out the linear position of the hi hat.

The circuit is rather confusing. Why you would tie an open drain high using the output of an op amp is simply beyond me. It's hard to believe that anyone would want to handle a switch in this fashion, even if you need to do inversion and/or level shifting. I suppose its possible that this funky circuit forms some sort of oscillator that when low pass filtered gives you a distance measurement, perhaps explaining the capacitor, but we'd really need to know what is being used as the output of this circuit to know.

Is it possible that the ORIGINAL original circuit used some sort of ratiometric Hall sensor, and not a switch?