OpenStreetMap keeps a collection of tags for each node, including "official names," so in that sense it does include records of these arbitrary renamings, but there are rules about naming things and to solve disputes/edit wars[1] (most commonly, the "on the ground" rule that reflects real world usage[2]). For example, see the Gulf of Mexico node[3], its wiki page[4], and the discussion following the executive order to rename it[5].
[1]: https://wiki.openstreetmap.org/wiki/Disputes
[2]: https://wiki.openstreetmap.org/wiki/Good_practice#Map_what's...
[3]: https://www.openstreetmap.org/node/305639190/history/143
[4]: https://wiki.openstreetmap.org/wiki/Gulf_of_Mexico
[5]: https://community.openstreetmap.org/t/gulf-of-america-gulf-o...