Per the link in your screenshot OSM is following the standard of using the GNIS official name, while also keeping the displayed name as the common name until and if common usage on both sides of the border begins to reflect the official name in GNIS.
Precedent from the Gulf of Mexico/“Gulf of America” discussion as well as accepted OSM convention would suggest that once GNIS is updated (which it appears it is already), official_name:en-US=* should be added/changed to Lake America, but name=*left as is unless and until common usage on the ground on both sides of the border shifts to prefer the “Lake America” name (which seems very unlikely).
Why are they a rube?
Per the link in your screenshot OSM is following the standard of using the GNIS official name, while also keeping the displayed name as the common name until and if common usage on both sides of the border begins to reflect the official name in GNIS.
Note: emphasis is mine