Quadrature demodulation, homodyne detection, lock-in detection - what's the difference?
They are essentially the same but may have key differences varying across implementation. The process is identical in terms of multiplying a signal by a sine wave at the center frequency of the signal (the carrier) which moves it to DC, and then low pass filtering that signal to maximize the SNR for the detection (presence of the signal at the carrier). 

For the differences that may exist the homodyne and lock-in amplifier may or may not be done with quadrature demodulation. Note the following from the relationship of multiplying two sinusoids:

$$\cos(\alpha)\cos(\beta) = \tfrac{1}{2}\cos(\alpha+\beta) + \tfrac{1}{2}\cos(\alpha-\beta)$$

If we multiply two signals of the same frequency in phase and low pass filter to remove the higher frequency term (as denoted by the LPF[]) we get:

$$\mathrm{LPF}\big\{\cos(\omega t)\cos(\omega t + 0°)\big\} = \tfrac{1}{2}\cos(0) =\tfrac{1}{2}$$

but if in quadrature we would get:   

$$\mathrm{LPF}\big\{\cos(\omega t)\cos(\omega t + 90°)\big\} = \tfrac{1}{2}\cos(90°) = 0$$

So a key question is what are you trying to detect? When trying to detect phase, the quadrature demodulator would actually be best as it would have the highest sensitivity to phase changes (the slope of the output voltage vs phase shift of the two signals being multiplied is highest when the signals are in quadrature). Similarly it will detect with maximum amplitude the presence of a signal that is also in quadrature (as this would be the first case essentially if both signals are at 90°). For arbitrary detection of any angle you would use an IQ demodulator that would multiply the input signal by cosine and sine with I and Q outputs each low pass filtered to show the detection for any phase of the input.