From what I understand is that by using analogRead, I will get the peak value from the sensor.
What leads you to believe that? Can you provide a link to where you saw that?
The analog output does not give the peak amplitude, just the instantaneous amplitude. To get a peak amplitude take many samples and look for the peak with software. You get ransom looking data because you are sampling the wave at different points on the wave. If you sample at exactly the frequency of the wave you will get a steady reading, but the reading could be any point on the wave. See aliasing.
The digital signal changes state based on a threshold that you set with the pot (not peak either).