Usually this only happens if you have 9 and 10 both installed.
Also there may be some Reader 9 preferences still in your Library.
In Home\Library\Preferences\Adobe look for any Reader 9 preferences or .plist files and delete them.
Also in Library\Support\Adobe you should fince some too.
I miss my Mac at this point because I can't think of the other places you would need to look.
If you find that Reader 9 is still in your Applications, download a trial of AppDelete. Install and open it. Drag the Reader 9 to the window and then move all the files to the trash.
Come to think of it, you could also try that with Reader X and then do a fresh reinstall. It should stop prompting for updates at that point.
I don't see any obvious .plist files for Reader 9. Just a few that don't specify which version they are for.
But your theory is basically correct. When the update notices originally started, the install would run but then report that it failed. Adobe's recommended corrective action was to manually download the update and install it. When I visited Adobe's download site, I noticed that the latest version of Reader for the Mac was 10.1, so I downloaded and installed it. A few days later, I again received a notice that I needed to upgrade to 9.4.5. After a few iterations of this, I searched the hard drive on my Mac and found the Reader 9 .app file and deleted it. Unfortunately, that didn't stop the bogus upgrade notices.
If you use AppDelete, and remove Reader X, it will also remove the associated support files from the library, my best guess is, including those that keep prompting for Reader 9 to be updated.
Once I found AppDelete, I quit just dragging apps to the trash to remove them, because of the long list of files that remained behind when I did that.