Are you saying that I would download an older version from that website (which is legal because the program itself is free but the ability to use the program is paid for through Creative Cloud) and then I authorise my use of the program by signing into my Abode account when the program opens? But on older versions, I may not be able to sign in to authorise my use of the program? Is that what you mean?
You need to install two things: the Creative Cloud app, which controls your license and then Illustrator itself. Some older installers for Illustrator are available through prodesigntools. And actually pdt just provides links to those installers, you do actually download them from the Adobe website.
And yes, I assume that you have a valid CC license. Without it, this all wouldn't work at all.