The thing that was confusing me about this was the five connections, because, like Branadic mentioned, there are three connections for a capacitive proximity sensor. Obviously, the heart of the transducer are these three components, but I was curious about what else was going on in there that would necessitate five connections. A first guess might be that pins 1 and 2 would be the center electrode and the guard, respectively, and the cable shield (labeled pin 5) is earth. Pins 3 and 4 could just be the two terminals to a diode used as a temperature sensor.
I used a sharpened probe tip to pierce the thin layer of epoxy covering the electrodes on one of the sensors, and I updated the schematic to show what I could surmise of the probe structure from doing this. There are at least 3 PN junctions in there, which I have represented as diodes. The guard electrode has a DC connection to Pin 1 of the connector, which seems a bit odd, as that pin drives the rectifier. The center electrode is attached to the common terminal of a pair of series diodes. When I simulated this arrangement in LTSpice, it reproduced the behavior of the circuit in a general sense. The servo wasn't railed, the output went more negative with larger capacitance at the sensor, etc. Pins 3 and 4 don't seem to have any connection (besides parasitic capacitance) to Pins 1 and 2, so they are probably a temperature compensation diode.
I will say, this sensor/probe combo is not great on the TC front, and I could not adjust the TC pot to get a TC on the other side of zero, or, in fact, one that was discernibly different from what it was when I started, which was on the order of 1 um/K. The copper shim I used was about 400 um thick to get the gap right only accounts for about 0.7% of this with a linear CTE of 17 ppm/K.
One of the reasons I am interested in this is that the CDCs I am aware of for such an application don't have the bandwidth to be of much use in spindle metrology. The AD7747 has a maximum ODR of about 45 SPS, and the full scale range of 8 pF is a bit too high to be useful. To be centered in that range with a 3 mm diameter circular electrode, you would need a gap of about 15 um, and there would be a fair amount of current to the target at such a high capacitance anyways, especially considering the guard will add about the same amount of capacitance again. Looking at Lion Precision's catalog, a "fine" resolution 2 mm probe would have a mid range gap of 100 um, or about 278 fF capacitance to ground, and the RMS resolution is about 0.85 nm (2.3 aF near midscale) at 10 kHz. So the white noise density is better than 0.023 aF/sqrt Hz (it's about 0.019 aF/sqrt Hz comparing the 1 kHz and 10 kHz resolution), or about 3 orders of magnitude better than the AD7747. Even this old unit I picked up does better than 0.1 aF/sqrt Hz white noise density.
MicroEpsilon makes the triaxial type, which is the type of transducer Branadic described. These have a much wider temperature range (ca. -273 to +200 C) than LP's sensors, which are specified from 4-50C, as you would expect from something that is just an air dielectric capacitor. These aren't too far off the Lion Precision specs. A transducer with a 50 um range, like the one I described above has a 1 nm RMS resolution at 8.5 kHz. They don't list the near gap, but the sensor diameter there is 1.13 mm, so if the near gap:measurement range ratio is about what LP uses, the white noise density in capacitance units is comparable. The probe drift for these two cases is comparable, at 40 nm/K and 60 nm/K for LP and ME, respectively.