Let's see the difference between mixing and modulating. The green wave is AM modulated and the blue is simply mixed. What is the 'real' difference? The blue is an audio reference with a RF riding on it. Just 2 signals. The green is an audio, RF, sum and difference signal. Four signals. So the question is how to turn 2 into 4 and then turn 4 back into 2?
The AM varies as the sum and difference of the 4 signals as you can see. In the mix it varies as the sum and difference of the 2 as you can see the RF ride the AF. Only 2 frequencies in the mix.
The full view. What makes the difference?
Now for the non-linear or distorted part. The most non-linear device in my parts kit would probably be a diode so insert one an look at that. It takes a few cycles to charge the circuit and then it resembles the AM signal. If that is true we should have the sum and difference. Let's take the capacitor out and see what happens.
The capacitor effect the circuit resonance. Adjusting the capacitor will improve the signal response. It bypasses RF to ground. It charges when the diode conducts and discharges when the diode is off. This action restores the missing half wave.
The cap sends RF to ground. With the coil disconnected the AF passes to the output.
Adding a diode and looking at the current we can see the pulse trains that could drive the output.
Adjusting the parts we can separate the AF or RF.
Changing the components produces the modulated RF signal. So the none linear device and filters will define the circuit.