Excellent post, I like the breakdown. So let's take a simple example for clarification:

"Why do race cars sound high-pitched when they are coming toward you and low pitched when they're heading away?"

Level 0: "That's because of the Doppler effect."

Level 1: "The frequency is f = (v + vr)/(v+vs) * f0 . Make sure you get the signs right."

Level 2: "Sound waves have a wavelength, but the wavelength is shortened when the car is coming toward you, because the car 'catches up' with its own waves, and lengthened when it's heading away. Since the waves travel at a constant speed in the medium, a short wavelength implies a high frequency, and frequency is what we hear as pitch. Yes, I know the equation. The phenomenon also happens for light, although the equation is different."

Level 3: "No, I don't remember the equation. Here, gimme a minute and I'll derive it."