There is also a description of the national Board of Geographic Names (BGN) who review and approve all names to be used with state and federal government maps and publications. Local governments and the array of organiztions are exempt from mandatory compliance but many comply for already approved names, only using their names for places not approved.
Also, individuals for their use, often with Websites, eg. waterfalls and lakes, or hking or nature guides use far more names than on maps, which is why where appropriate my Web pages clearly state I restrict names to those on USGS maps or approved by the BGN because you can research and reference them.
In addition, there is a process for individuals, organizations, government agencies, etc. to apply for official recognition, see above link, for inclusion in name registries, with maps and for publications. I'm currently researching one, Windy Knoll, a 1930's era firelook out campsite.
For Mt. Rainier NP there are currently 692 names for places, just over double from the 334 placenames on the first map of the NP produced by the USGS in 1915, see history of USGS maps for Mt. Rainier NP. Not all the names are on the USGS maps, including the latest NP map (1971) or the 7 1/2 minute series maps (1971-2000).
Anyway, that's it and now you know if a name for a place is official. Or not.