The first picture can be connected to an analog input, and the Arduino should detect a change of the analog value.
In the second picture the fet is used to interface with the microcontroller.
That way some kind of open collector compatible output is created.
But the Schmitt-trigger output of the second 40106 gate is already a digital signal. You can tie that output to the Arduino pin, without the fet.
The Arduino sketch should be a frequency counter, which is a lot harder than the first picture.
Since you don't want to detect a coin at 5 meters deep, the frequency counter solution should work for you.
The frequency range of a metal detector could start at 1kHz, up to the MW range 150kHz ... 1500kHz.
So if you make your circuit somewhere between 5kHz to 10kHz, that seems okay to me.
The frequency determines what kind of metal is detected best. With a metal detector that uses two coils (a transmitter and receiver coil), copper and silver is detected best with 3kHz and gold with 20khz to 70kHz. That is why the expensive metal detectors have multiple frequencies.