LR CC 2015 and LR 6 are the SAME PROGRAM. So updating shouldn’t change 6 to CC or CC to 6.
What does “Help / System Info” show for the version of LR you have?
What does http://www.adobe.com / Sign-in / Manage Account show for Plans & Products as well as View All / View Your Products?
And how much of what currency did you pay to Digital River? You may well have purchased a LR CC subscription although the cheapest option I know of also has PS CC as part of it which your screenshot shows is clearly not the case since it shows Try not Install next to PS CC 2015.5.
The Adobe Creative Cloud Desktop App, what is pictured in your first post and first screen shot, doesn't know how to display LR in any other way other than CC. Your System Info screen shot shows you are running LR 6 and not the CC version.
Both programs are exactly the same download and the exact same file installed on your hard drive. The difference comes from how you Sign In to the program. You are Signed In for the 6 version. In other words the Adobe ID you are using is not associated with any subscription plan.
You can remove that Creative Cloud Desktop app from your computer, it isn't needed. The use this link to check and download updates for LR.
Many thanks for this.
Everything runs smoothly once I have uninstalled the CC app.