Hi, and welcome to OpenStreetMap! Nice additions here - contributions like this are really valuable for hikers since a lot of external services draw from OSM for paths and stuff.
Generally, the name tag is for the official or "common" name of a feature. That means it might appear on a signpost or map board or something. If not, you might consider using other related tags like `loc_name` (for informal names that a local might use casually) or `destination`. (see https://wiki.openstreetmap.org/wiki/Key:destination)

Let me know if you have any input or questions. I haven't modified any of the stuff you added, just want to get your thoughts. Thanks again for your work here, and welcome to OSM!