Okay. Not trying to be a stick in the mud, but I know you can do Messages on a Mac, but cannot on a Windows computer. I am not exactly sure how one accesses Messages on the Mac, but I didn't think that it was by logging into iCloud on a browser. And I know that the Messages icon has never been in iCloud when I logged in on a browser, no matter what I am using for backup. But I have been using iCloud now for almost a year for backups, and it has never been there. When you are backing up in iCloud or iTunes, it is one file, and nothing is separated to go somewhere else, such as to a Messages access.
I guess this is more for both of us to clarify what works, since I don't know Mac, and I use Windows.