There are many widespread approved tags which can be added to Trails to reflect the private, or discouraged nature of trails:

https://wiki.openstreetmap.org/wiki/Key:trail_visibility

https://wiki.openstreetmap.org/wiki/Key:trailblazed:visibility

https://wiki.openstreetmap.org/wiki/Key:smoothness/Gallery

https://wiki.openstreetmap.org/wiki/Key:sac_scale

https://wiki.openstreetmap.org/wiki/Tag:informal%3Dyes