Which I get. But clicking on that link you provided does not give you the option of choosing between 32 and 64 bit installers.
Yeah, that was certainly a 32-bit installer file being referred to in the other message, too. (The 64-bit installer file is called iTunes64Setup.exe.)
Just in case it's an issue with your normal browser not working properly with the system-detection equipment on the page (there's a javascript control that checks for operating system specifics, and is supposed to deliver the correct Mac or Windows installers), let's try a different web browser for the download. If you use Firefox instead of IE (or vice versa), do you get offered an iTunes64Setup.exe instead of an iTunesSetup.exe?