Several trouble-free thousand miles since my last post, I finally dismantled my RD by following rc300's excellent photo-instructions.
I was surprised to find my Philips-headed stop-post was only finger tight. Good job it didn't fall out last week when I was on a 1000km Asian tour on the bike. (I painted nail varnish on the thread to "lock" it and I'll keep an eye on it in future. Ditto the little Philips screw that holds the arm to the body.)
The photos here are of the near identical Shimano XT, but I found the Lx +not+ to have any "O" ring; instead it has a 3mm wide, 1mm deep, channel in the pivot shaft. I presume this channel is filled with grease when new, but mine was bone dry.
I got, and tried to fit, a couple of small "O" rings side by side in the channel, they fit snug, but when I pushed the shaft into the body, they were merely forced down the shaft instead of sitting in the channel, so I abandoned that idea and just filled it with thick sticky farm grease.
I also made a new plastic shim from a telephone charge card to take up slack under the plastic collar at the base of the shaft.
I cleaned, lubricated, and reassembled it all, adjusted the gears, and it works fine.
The lateral play I originally mentioned is no more than 5mm side to side now, the same as my new Sora road-bike RD. And that play comes from the parallelogram body.
I should mention that it is not necessary to remove the RD from the frame to do this, just take the chain and wheel off.
So what looked daunting, turned out to be easy, and well worthwhile, thanks to rc300.
This wonderful old bike is now entering it's 18th year of service, and it is still strong and sweet as a nut.