Thanks!  That's very helpful. I will start applying what you are marking as names for these trails.

Regarding OSM, names need not be official, but they shouldn't be used to describe things. 
https://wiki.openstreetmap.org/wiki/Good_practice#Don't_use_name_tag_to_describe_things

We strive to represent how the community or signage on the ground represents something, which seems to be the situation you describe.