Well, I think I finally figured it out, although there is still a mystery.
The problem has been that I could not cut a 1.25" diameter mild steel bar off in the 3-jaw chuck without house-shaking chatter. This with a HSS T-blade 1/8" wide in an Aloris BXA-7 holder.
I tried many things, mostly to rule out one possibility or another that came to mind. The gibs are now all tight, and so on. The Aloris BXA toolpost replaced the warped Dickson that came with the lathe
On the off chance that loose back hold-down plates on the carriage were the issue, I used a C-clamp and a piece of aluminum (to protect the bed way) to clamp carriage firmly to bedway. No effect.
Running fast and slow using the variable speed drive and/or the VFD had some effect, but gross chatter happened even at very low speeds.
I recently bought a used slotting tool bit, consisting of a 1/16" wide carbide blade brazed to a 3/8" square shank, the blade protruding about1/4". Even this chattered.
Huh? How does that work? We are making a tiny groove using this 1000# machine. This ought to be easy. Time for some reading or re-reading. Marlow suggested cutting off (parting off) at one third the speed used for turning, and pressing hard if it chattered. This worked for the little slotting tool, although I did manage to break it by pressing too hard. Well, the used bit cost me $0.75, so I happily shed 3/4 of a tear. Clearly, we have progress here.
So, why then did the tiny slotting tool chatter no matter the speed? Something must not be stiff enough. But what? I've tightened or clamped or adjusted just about everything, to no avail.
I woke up the next day with the answer -- torque. I was going slow to be sure, but was not using the back gear, and so the drive system was not torsionally rigid enough. When I used the back gear, the chatter went away, and I was able to part that 1.25" diameter bar off without danger of shaking the house apart.
What also seemed to help was that I was using a mister to spray lots of Rustlick WS-5050 emulsion right into the bottom of the deep groove. Trying to keep the cut lubricated using a hand brush just was not working, especially when the groove became deeper than it was wide. However, coolant by itself did not abolish the chatter. The back gear is essential.
But, I did notice the whole toolpost and holder tilting when I leaned into the cut, so I'll have to track down if something is still loose and needs to be adjusted. Or perhaps it's normal, given the forces invloved.
The other thing that happened is that the toolpost and/or compound rotated perhaps 5 degrees under the stress of parting off, causing the blade to drift out of perpendicular with the bar being parted. This caused a lot of trouble until noticed and fixed.
One thing I noticed when I first got the lathe is that one of the two5/16-18 swivel bolts (055-017) that lock the compound against rotation had been stretched enough to visibly distort the threads, and I always wondered why someone would apply that much force. Perhaps this creeping is the reason. (The stretched bolt was replaced, and the newer bolts are slightly beefier to the eye and may have been hardened.)