Your basic understanding of the underlying electronic principles is correct: The VOR receiver detects a phase offset between two signals, which tells the receiver what radial the aircraft is currently on. Wikipedia has a decent explanation, and this article on Digital Flight Instructor has a nice widget you can play with to see the frequency offset.
The underlying electronic principles the receiver uses are generally not useful for pilots: if your VOR receiver dies you're not going to hook up an amplifier to an oscilloscope and try to sort out the phase offset yourself, if for no other reason than your head would be buried in the cockpit staring at the oscilloscope as opposed to looking out the window for traffic or watching your flight instruments.
In addition, the "strobe and sweep" explanation is generally easier for students to understand: Imagine the VOR station has two lights, a white light that rotates, and a red strobe that fires ever time the white light passes the 0 degree (North) radial. By counting the time between when you see the red strobe and when the white light passes you its possible to determine what radial you're on.
The mechanism by which the needle is deflected depends on the type of indicator, but it basically boils down to "The receiver does some magic and converts it into a number of dots of deflection on the indicator". The deflection can displayed by moving needle attached to a meter (applying a specific positive or negative voltage), blinking some LEDs, or showing a graphical representation of the CDI (e.g. on a GPS display) with a mark over the appropriate dot.
How this works for each specific indicator is generally only of interest to an avionics technician, and the information would be contained in the service manual for the radio and indicator.