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The title should probably be changed to "Which Linux distros are the best?" because it does not even attempt to identify a single 'best' Linux distro.
Anyway, here is my take. (Disclaimer: I am a Gentoo user, so my view may be biased)
By far the most popular distro is Android. This year, for the first time Android will overtake Windows as the most sold operating system (installed base is projected to pass Windows in 2016).
A distant second is the OpenWrt/DD-WRT family of distros.
Most User Friendly
I'll give that title to ChromeOS. Nothing for the user to care about, even his data will be backed up into the cloud automagically.
Ubuntu gets second place from me, it is not quite as polished, but I think its Unity desktop is now more user friendly than Maté or Cinnamon.
Best Live Distro
That's a tougher one. A live distro to rescue your system? Systemrescuecd. A user friendly live distro to do productive work? Ubuntu.
Best for Linux Connoisseurs
I agree that Arch is a good choice. I'd vote for Gentoo (if you don't mind the compiling) or Aptosid (if you do) though.
Best for Learning Linux In Depth
Gentoo is excellent if you want to learn Linux and then continue to use the system for daily work.
However, LFS should at least be mentioned here. The learning curve is steeper. But it is difficult to keep the system updated and working at the same time.
Best for Older Computers
I'd say Debian. Arch might fit the bill too, but has no support for pre-i686 machines.
Best Office Desktop
All desktop distros come with LibreOffice so there is not much difference there. Maybe if it had said "Best Enterprise Desktop" a winner could be found.
Most Stable Across Releases
Of course enterprise distros offer enterprise-class support. But if you are only looking for end user support, the value/$ quickly deteriorates with Red Hat and Oracle.
Some categories which were not mentioned in the article:
Debian sid is the classic choice. Aptosid makes it a bit more palatable. Arch and Gentoo also do very well here. Fedora will have the kernel and desktop-y parts up to date at least.
Gentoo and LFS.
Best software selection