It turns out that the galvos I bought are closed loop devices. When a signal is applied to them, the move to a new position, and they also give feedback about their position to their controller. This feedback "closes the loop".
Taking the rear cover off one of the galvos shows the position sensor. It consists of an LED, two photodiodes, and a plate attached to the galvanometer shaft. As the shaft rotates, it allows more or less light to shine on each photodiode, which provides the feedback.
I had been wondering why the galvo needed a six-wire connector. Now it seems obvious that two of the wires are for the galvanometer itself, and four of the wires are for the position sensor - V+, V- and two photodiode signals.
This complexity also explains why each galvo control board has 4 quad op-amps: the board is an analog computer, trying to calculate how to accelerate and deaccelerate in order to bring the mirror smoothly to its new position with no over or undershoot.
I learned about the galvo position sensor from zenodilodon's Youtube channel, in particular the video "Inside the Closed Loop Laser Beam Stearing Galvanometer" in which Zeno shows several galvanometers that look like bigger versions of my cheapy.
Yes, the production is a bit rough. Yes, those dead pixels are on Zeno's camera sensor, not on your screen. Yes, I too find the paint flecks on his hands distracting. Despite all that, it's a pleasure to watch a gifted professional dispensing the kind of helpful information and tips that can only come from experience. I am totally subscribed.
Another one of Zeno's videos, "Complete Build of a 4 Watt Analog Modulated ILDA Show Laser" was also helpful. I'm building a just-good-enough for home single color, 5mW , digital version of the professional RGB, 4 Watt, four laser analog rig that Zeno puts together here. Plenty of good tips. My build will be simpler and involve less swarf.