I solved the problem.
I deleted the old computer fro my profile on the old PC and added it back in.
In HP Support Assistant on both computers, I logged out and in and this appears to have fixed the problem. Note that originally, HP Support Assistant was logged in on both computers and to the same username, and the new login was to the same username.
I also added HP Support Assistant to my smartphone.
I found that on one of my computers, there were two devices for my old PC, one with a shortened product description and another with the full description. I removed the one with the shortened description.
On the smartphone, there were all sorts of printers listed, in particular printers I had stopped using from a few years to 13 years, and which I had removed from HP Support Assistant on my older computers and/or not added to my newer computers.
There is also an entry on my smartphone for a device that's really not a device. One of the devices has a product number which is really a one year warranty extension for my old PC (which I did get).