How does the accelerometer in DE0-Nano work (in words)?

The equations are correct as such (seriously speaking, I didn't check them in detail), but they don't describe the operation principle of the sensor.

In simple words, you're assuming that a movement is sensed by a generated displacement current. But that's not true, instead the position of the mass is sensed by a differential capacitance sensor. Both quantities are different by one d/dt operation, very important for the sensor's capability of sensing a static acceleration.