Well for anyone who finds this thread in the future, here's what I did:
NOTE: I read through the documentation of the Cleaner tool from Adobe and I swear it said to uninstall the program before running the Cleaner.
I wouldn't advise uninstalling the program before running the Cleaner because you can only run the tool on Adobe programs that are currently installed. So yeah, major fail there on Adobe's part, genius programming and implementation. I have no idea what they were thinking.
1) Plugged in the old hard drive. [E:\]
2) Uninstalled Illustrator from Add/Remove Programs (Windows 7) [as advised by Cleaner instructions]
3) Restarted computer [as advised by Cleaner instructions]
4) Ran the installer for Illustrator that I'd downloaded from http://prodesigntools.com/adobe-cs6-direct-download-links.html
5) Restarted computer
Everything works fine now; I confirmed by removing E:\ drive and running the Updater, which successfully updated Illustrator.