OSM Carto renders name= over official_name= since generally official names are a lot longer, an example being Carto shows Deutschland for Germany instead of Bundesrepublik Deutschland; for most purposes it's a lot smoother. 

Generally, changes can take a fair bit to render on the map.

Gulf of Mexico/America still shows as Mexico because that is in the name= tag, and this was agreed upon via consensus, see here: https://wiki.openstreetmap.org/wiki/Gulf_of_Mexico

I do not know whether Denali's name= should be put as Mount McKinley or kept as Denali; that might need more consideration from the rest of the community. I personally think it should be kept as Denali (with official_name being Mount McKinley) because that's what most people call it, however i'm unsure how the greater OSM community feels.