So Lightroom is telling me that there's is a new update available with Facial Recognition, HDR Merge and Panorama Merge. When I click the download button, it just takes me to the Lightroom CC webpage and does not show me where I need to go to download the update.
Where do I download the update?
Currently, the stand-alone and the CC is the same download and installer. You can later license as CC or as stand-alone (as upgrade also).
There is no update, it's a sales pitch for Lr 6
Adobe call this an update.
No, it's not an update – it's a full product upgrade, and will install side-by-side with any older versions of Lightroom that you have, including LR5... You can uninstall the prior release afterwards if you no longer want/need it.
This upgrade is free/included for CC subscribers (Photography Plan, Complete, or Student), or is a paid upgrade/product for standalone/perpetual customers.
As for where to download it, if you aren't seeing it easily available then you can just use the Lightroom 6/CC direct download links.
Hope that helps!