what is the distinction between /media, /mnt and /run/mount?

FHS v2.3 was released ten years ago. Some things have changed since then (including the introduction of /run¹). About three years ago, the Linux Foundation decided to update the standard and invited all interested parties to participate.
You can view the v. 3.0 drafts here and the section that describes /run here.
The distinction between /media and /mnt is pretty clear in the FHS (see Purpose and Rationale), so I won't go over it again. Same for the purpose of /run - see links.

The Gnome story is yet another thing. Gnome uses underneath an application called udisks (replaced later by udisks2) to automount drives/devices. For quite a long time, udisks default mounts were under /media. In 2012 the devs decide to move the mounts to /run/media (i.e. a private directory). So the different behaviour you're experiencing there is caused by the different versions of udisks that each DE is using.

^{1: see
What's this /run directory doing on my system and where does it come from ?
What is this new /run filesystem ?}