Final, perhaps, update. But, there's still an open puzzle that someone might recognize.
Bottom line, I think stuff is working. Super hot weather combined with the presence of the Moon have "prevented" me from doing any additional imaging, but I have looked at the befores and afters and think I might understand what was going on, though not why it got into that state. The key parameter seems to be the mount's reported guide speed, which changed by a factor of 100 somewhere along the line. The open puzzle is that it seems to have been wrong initially (on the system that worked), and got "fixed" after. Whether that fix was in the software update I did, or a side-effect of some guiding experiments, is probably lost to history.
I have compared the four PHD2 configuration screens (global, camera, guiding, algorithms) from the working system back in May, to the non-working one in June, and to what appears to be working now. All the same. But looking at the calibration data, there are differences.
Below are 4 screen shots of PHD2's calibration screen. The first is the legacy working system. Note the RA and Dec guide speeds are about 50x, which seems oddly large. Bringing up the mount's configuration utility on the hand controller, it says 50%, not 50x. But this system calibrated very well, as can be seen by the graph on the left.
After the software update, and after a recalibration, but *before* the mount and configuration resets is the June graph below. This is the state of things earlier in this thread, where the mount was still running off in the Dec direction, apparently on its own. Note the guide speeds have dropped by a factor of 100. Also note the faint string of dots on the Dec line, which I've not seen before. I remember a lot of fussing about backlash during the calibration, so perhaps that's what it is? But the puzzle is why the guide speeds changed so dramatically. I do recall trying to adjust the RA speed over the past few months, in an attempt to improve the RA guiding. But that didn't help, and I returned the rate back to where it was before. Never touched Dec, as it's been great. I think the changes were done on the mount hand controller, and without a recalibration, so this might have confused the software. But both RA and Dec numbers changed in scale together.
The next experiment, again without resetting the mount or parameters, was to roll back to the backed-up May SD card image, to see if I could simply roll back the clock to a happier time. What I found was that PDH2 still complained about the guide rates being different, and that I should recalibrate. That I did, resulting in the calibration data below. Note the extremely large spacing between the dots; this was accompanied by a complaint of a large step size, but I accepted the calibration and got some good image data that night. Note here the guide rates are back to 50x, though I had not changed the mount's settings. (Ignore the Created date here - the Raspberry Pi didn't have an Internet connection at the time, so had no idea what date it was.)
Tabbing forward. Thinking about the rates and such, I did the factory reset of the mount, and found the reset button on the Advanced tab under PHD2's Guiding configuration screen. Re-calibrated yet again, and got the last (and current) calibration data. Again we're back to 0.5x guiding speeds, and a reasonable step size between dots, though not the same as before. I think this is probably correct, as I calibrated a lot closer to the celestial equator this time. No faint dots off the end of the Dec line, either.
SO... where am I? As noted above, I have only had the one imaging session with the current guiding parameters and calibration. It wasn't a spectacularly good guiding session, from a numbers perspective, but not unusable. Along the way I have fussed a bit with the telescope's balance on the mount, and may have lost the sweet spot that I had been running with before. The mount's bearings are rather stiff, so knowing where balance actually is can be annoyingly vague.
If anyone has any idea of what might have led to the 100x difference in interpreting the guiding speeds, I'd appreciate knowing what it was. Was there a software change in interpreting the mount's guide speed? If so, where?
As things are currently configured, I believe I have a usable system. Just need a good night, sans Moon and heat, to verify that.
Final random thought, putting on my software engineer hat for the moment, and looping back to the thread's original topic. Why was the mount running off in the first place? The only thing I can think of is that with the speed factors so out of whack, perhaps the software ran off the end of some signed integer arithmetic, wrapping a positive number to negative (or vice versa), making a corrective nudge into one that made things worse. Evidence for this is that there were some initial nudges in the right direction, then an accelerating deviation off the top of the screen, in the presence of "corrections" from PHD2 that should have had the reverse effect. Seems like there was a sign flip in there. I haven't looked at the source - not even sure which source to look at, PHD2, INDI, or the hand controller - but it kind of fits.