Thanks for sharing! Nice document, it is very good to be aware of the fact that your lock-in amplifier does not just directly measure dI/dV.
We don’t do DC in my group anymore, but i used lockins myself extensively in my PhD for capacitance measurements (at “pretty” high frequencies of 100 kHz and up, at least “pretty high” for a lockin 
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I do remember finding this document from SRS a useful introduction to what a lockin is and how it works BTW:
https://www.thinksrs.com/downloads/pdfs/applicationnotes/AboutLIAs.pdf
In particular, in my phd, as I was working at higher frequencies, the phase lag effects you mention become even more important (since they get much more difficult to neglect…) so I had to take the bull by the horns and really understand my phase shifts. (Figure 2-8 of my thesis below might be useful for a starter on this.)
My general advice is to never look at only the amplitude R of the lockin. Ever. ALWAYS look at BOTH quadratures X and Y. The reason is that if you have a stray crosstalk signal coming from a capacitor, it will appear (in the simplest case) in the orthogonal quadrature from the conductance signal that you are after. If you look at only R and do not pay attention to what is happening in BOTH X and Y, then any capacitive crosstalk signal will screw up your interpretation entirely since it is adding in quadrature! However, for relatively simple networks, there is a correct “reference phase” you can chose for which, for example, the capacitive cross talk is only in Y and the conductance signal would be only in X.
But improtant: this is NOT the X defined by pushing the “auto phase button”, since this picks the phase that puts the current reading of the quadrature combination of the crosstalk and conductance signals!!! You have to somehow “calibrate” what the correct reference phase is that makes X (or Y if you want) into your conductance signal.
I did a lot of this reference phase calibration in my PhD:
https://nsweb.tn.tudelft.nl/~gsteele/thesis/Gary_Steele.compressed_images.pdf
In my experiments, I could change the capacitance in my circuit by moving the tip, and thereby calibrate out the phase angle at which the capacitance signal was oriented. Setting my reference angle based on this, I could then just read out capacitance from X and the equivalent resistance from Y.
In principle, if you have a gate that changes only the conductance of your device and not any relevant (pF / nF) capacitors that are relevant for your <100 Hz phase shifts, then you could play a similar trick: for a given configuration of the filters in the setup (including any outside your fridge!), you could change the conductance of the gate a bit and see which quadrature the conductance is appearing in for that frequency and configuration of the setup. You can then check your assumption by changing the gate voltage and seeing what happens.
Now, of course, if you record both X and Y in your datafile (or R and theta, but X&Y are better), then you can always POSTCORRECT your reference phase choice by just performing a phase rotation of the signal in software later. If you have only a single point, then this is useless. However, if you have an IV, or a gate trace, or a 2D image map, then usually there is a “correct” choice of the reference phase that shows you only conductance in one of the quadratures and produces a featureless second trace / image related to the (constant, device insensitive) capacitive crosstalk that gives you leakage into the other quadrature.
(There is even a special version of spyview I wrote called “spyrotate” that opens up two 2D quadrature images and gives you a slider to rotate the reference phase until you get it right! I used the purely capacitive signal from a gate on the 2D at zero field for my capacitive phase reference. Once I saw that the gate was no longer visible in one of the images, I knew that I had found the correct reference phase. It would be mega easy these days in a jupyter notebook with ipywidgets)
(Also note that if you have big enough inductors in your circuit, you could also have a LC crosstalk signal that would give you a fixed static offset of your conductance quadrature! It would require unusually large inductors though to give a relevant phase shift at < 100 Hz…)
Some differences with conductance of quantum transport devices are that the conductance of quantum dots for example can vary by orders of magnitude. If your device itself is loading a capacitance that is giving a relevant phase shift, then the reference phase at which your conductance comes through depends on the conductance, and then this no longer really solvable without a decent independent estimate of the capacitance that is forming the filter with your device. I think though that this type of problem is avoidable if you’re careful in how you set things up. And also my personal experience (in both capacitance, but also “dc” lockin measurements in my postdoc) is that fixed (ie non-device-dependent) crosstalk is a bigger problem. And finally, once you understand how to calibrate the reference phase correctly, it is an easy task to check if this is the case for your device / setup by recalibrating the reference phase for different conductances of your device and checking if the phase is the same.
That was my experience at least, though it might be nice to share some of my knowledge as a “lockin veteran”
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. But yes we should do that after a couple rounds of quick comments from the experts.