After further investigation, it appears that the display driver within the phone itself is faulty. I put the display back on the original phone and it works fine.
That original phone, which was acting crazy when buttons were pressed, now works just fine since I loosened the screws on the main board just a bit. This workaround prevents the buttons from being pressed too hard, and voila, fully functional phone.
Still, I'm left with a "headless" M64, with the nice Executone lady telling me the the button is set "as shown on the display". (At least they both work now though.)
As for the other one I mentioned above in this topic, it seems that we just have bad handsets, dropped on their heads too many times. Would any of you know where I might be able to get some (2~3) of these at a reasonable price? (I don't mind if they are Executone brand or not.)