inductive sensor (inductive limit switch search terms) but I think they will trigger at different distances based on bit shape/size
exactly. Inductive sensor repeatbility is very poor, because depends on many factors including just aging.
Touch to close electrical loop at about 5mA with about 10V is proven industrial method.
Sure, there is a good reason to use Arduino for double or even tripple touch - first to determine where the sensor is, roughly, plunging the plate down. Two more with slow feedrate for accurate sense. And generate signal to control "Zero is here". It's also good to use less than 2V for final touch.