Yes, I did register and log in. In the meantime, I decided to download VMLiteWorkstationSetup.exe, which is available on the downloads page. This has been successful. After I downloaded and installed it, it created a Windows XP Mode VM very easily, basing it on the XP vhd that Windows XP Mode had created. Perhaps this was what I was supposed to do all along; in fact, I just tried again (after a reboot), and this time, to my surprise, clicking the download button on the page vmlite.com/index.php/products/vmlite-xp-mode/download
does actually start to download VMLiteWorkstationSetup.exe. So maybe it was just a temporary problem with cookies or something earlier.
Still, the way the site is set up, VMLite XP Mode looks like something different from VMLite Workstation. It's not clear that VMLite XP Mode is what you get when you download VMLite Workstation and let it import Windows XP Mode.