The first thing I would check would be the /help/about dialog to make sure you installed version 4, which I really doubt since version 4 is two full versions old.
I think you have three choices-
1. Download LR-4.4.1 and install with your LR-4 Serial Key. Adobe - Lightroom : For Windows
2. Buy a 'Standalone' upgrade to LR-6/CC. You will need your LR-4 Key and the new LR-6 Key.
3. Buy a subscription to LR-CC.
Your LR-4 Key will only work for updates up to LR 4.4.1
OK, now I am impressed that all those old versions are actually available for download.
Thanks for the suggestions. But, when I use the link you suggested for the download, it still looks like it wants me to download LR-CC rather than just Lightroom 4.1.
Are you clicking on the v4.4.1 link?
This is what I see on the link page-
and when I click on "Adobe Photoshop Lightroom 4.4.1" I get to Adobe - Lightroom : For Windows : Adobe Photoshop Lightroom 4.4.1
which has this download button-
(This is the Windows version!)
Mac is here- Adobe - Lightroom : For Macintosh
If you still have trouble, then maybe the site links are specific for a country?
Try here also- Product updates