Some input (guesses) from my long electronics experience...
If you look at the four points, notice that there are a pair of circuit board traces to the big "disc", and another pair to the interleaved contacts on the inner "disc".
Any button contact closure uses 2 traces, and a pressure sensor or piezo element would also use two contacts.
It seems to me that the four points have two devices each.
The interleaved fingers on the inner disc a very typical design for buttons. A small voltage is present on one set of the fingers, and the other side goes to the devices "input". When the button is pressed, a tiny conductive pad is pushed onto the fingers, completing the path and sending the voltage to the device, signaling that it is pressed down. Tear open an old calculator and look at the board under the buttons and you will see the same fingers.
In the S2 pic, these buttons are covered with a bubble of soft plastic, i.e. bumpers. It seems the screen would actually rest on these. These would offer the feeling of a click as they are normal buttons.
The larger disc appears to support the button "fingers" above the button "shorting pad". The large disc is the mystery, but is likely some material between the two big metal discs. The big circuit traces likely go to the discs, to either supply voltage to them (to unlock, or soften the material between them), or sense voltage from them (sensing degree of pressure). In other words, the material between those discs probably has an electromechanical property, and forms the "springs" or "sponges" that support the display. Piezos are usually in that same form, but it's impossible to tell what they really are from the pics.
Piezos can sense teeeeny variations in pressure. If you have four under a rectangular panel, it would be simple to triangulate the position of at least two pressure points just by looking at the difference in pressures (voltages) from the 4 piezos. But then why are there traditional buttons as well?
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