Examples
This post contains the examples of the (sub-)categories above.
karrenweg
(single_track, multi_use, carless, etc.)
Trail, easy_trail, wide_trail
Trail, difficult_trail, narrow_trail
Presence of obstacles on the side makes this one a narrow path:
Trail, disputed whether easy_trail or difficult_trail
Trace
(the rock stairs are the trace + blue markers)
Off-Trail
“Pathless”, trackless, no visible traces or markers on a way but terrain suitable for crossing on foot.
MTB trail
(mtb_single_track), notice the constructed curved wall made specifically to easy going through the curve at speed on a mountain bike
A trail (easy_trail) next a MTB trail pictured above (on the right you can see the constructed bum described above`)
Other examples of mtb_single_track
A pedestrian would most comfortably walk along the red line. For a biker going downill, momentum makes the yellow line more logical and enjoyable (the difference is faint but discernible here).
Mountaineering
(only the first photo one has a license that makes it ok for the wiki)
Existing OSM ways that would be tagged with mountaineering