When you say "I also read that difference in phase does not mean less coherence at a given frequency", you are right. But "As I understand it, a shift in phase requires a shift in frequency" is not correct. There is a misunderstanding here.
As far as your exact question is concerned, "if I've taken the Fourier transform of two waves, and am looking to see how they are correlated in the frequency domain, what am I actually measuring?", it can be answered better without many equations. Since you are a biologist, it might be a better approach for you.
Correlation in frequency domain: There are two things to consider at each frequency, (1) amplitude and (2) phase.
First, when talking about normalized correlation as described by @Carlos, then amplitude does not play a role as it appears in the denominator as well. Remember that it does have an effect in correlation, just not in normalized correlation.
Secondly, if you have two signals X and Y, then spectral coherence tells us how their phases are related at each frequency. Suppose that for every f, if X has a phase of $\theta$ and Y also has a phase of $\theta$, then in their cross spectral density, $\theta$ cancels out because
- In multiplication of two signals in frequency domain, phases are added together.
- Conjugate in one of those two signals (see definition of cross spectral density) reverses $\theta$ to $-\theta$.
Then, their amplitudes are added in full coherence with all the energy in $I$ part and zero energy in $Q$ part. See the figure below.
For different phases $\theta_X$ and $\theta_Y$, this does not happen. To see this in concept in figures in more detail, you can read my article in full.