I suggest you check what System Info (Help menu) is indicating. On first line you should see the actual version installed. Lightroom 6.10.1 and CC 2015.10.1 is the most recent build,
It's saying the same thing: Lightroom version: 6.0 
Have you tried this link- Download Photoshop Lightroom
Click on the > and then scroll down for Mac & Windows updates.
Note: Do not un-install v6. V6 is the "Base" program install, and is needed for the 'Updates' to be applied.
I've tried DL and installing from the page that you linked to above. I first tried the 6.2 installer and then the 6.3 installer. Both installers ran and both ended with the comment window saying that the installation was successful. But the splash screen and menu LR->Abount Adobe LR both say I'm still on version 6.0 even after running both the 6.2 and 6.3 installers. So then for fun I tried DL and installing version 6.9 and when I ran that installer it finished by saying installation failed. I'm still on version 6.0.
Can anyone from Adobe help?
(sorry- not Adobe here, just another user)
Can I suggest the following (if you have not done so already)
1. Run the CC Cleaner Tool after reading all the info here Use the Creative Cloud Cleaner Tool to solve installation problems
The link for the Cleaner download is in Step 6
( I know you are using "Stand-alone" but do the clean steps anyway. I helps fix many install problems)
2. Use the link in Post#3 above to download the "Application Installer" for v6.0 - install.
3. Use the link in Post#3 above to download the v6.12 "Update" - install.