A final note -- I hope.
When I re-attached the old backup drive to do a secure data wipe, it appeared to interfere with the new backup disk, even though the old one was attached via a USB2.0 cable, and the new one on a 800 FireWire cable.
The "can't find the backup disk" error recurred, and I just let it be until the data wipe completed 30 hours later.
The old disk has been returned, and the new disk appears to be doing its job.
In checking some other things, I noticed that there were some old programs still in the automatic startup list (I found reference to them in the normally hidden system directory /Library/StartupItems
There were some old Norton anti-virus stuff, some old Virex stuff, an old LexMark printer driver startup -- all of these were left-overs that never got removed as I migrated my iMac hardware a couple of times and as I upgraded MacOS from Tiger to Mountain Lion.
I got rid of these after carefully checking that they were not required. Most were running the old PowerPC executable code and was not able to run on my newer Intel iMac (late 2010 vintage) under Lion and later.
There was also an old thing called "Retrospect", again not executable under Mountain Lion.
Sometimes I wonder how this old stuff that never gets fully removed hangs around, and if it can possibly interfere with the smooth operation of Mountain Lion.
I'll never know, I suppose.
As I have time I try to go through my Applications folder and test each app to look for obsolete stuff, and get rid of it.
Hopefully the story about the backup disk problem is over. I'll wait a few days and do one final update, unless the problem recurs. Oi.