To elaborate on my earlier example, this reservoir is located entirely in Ohio. However, locals and the state government, which manages the lake, insist on referring to by a compound name, while the federal government insists on a simplified name due to federal naming policy. Both names are in the same language but have different scopes, which are clarified by loc_name, reg_name, and nat_name, with name breaking the tie in favor of the local and state name.
Adding a border to the situation, this reservoir straddles the Georgia–South Carolina state line and is managed by the federal government. The reservoir was originally named Clarks Hill Lake. Later, at the federal level, it was officially renamed after Thurmond, as a birthday present from a South Carolina senator to a Georgia senator. However, the states are free to continue calling it whatever they want. Most people on both sides of the border still call it Clarks Hill Lake, which the federal government recognizes as an alternative name. We have that as the name, with Thurmond relegated to an alternative name.
Similarly, this reservoir straddling the North Carolina–Virginia state line is also managed by the federal government. It was originally named Bugg’s Island Lake (or Buggs Island Lake due to the federal rule against apostrophes), but Congress later renamed it the John H. Kerr Reservoir after the Congressman from North Carolina. North Carolinians call it Kerr Lake, but Virginians prefer its original name, having tried unsuccessfully to revert the federal name.
Based on @amapanda_ᚐᚋᚐᚅᚇᚐ’s example, I’ve added name:en-US-NC and name:en-US-VA to the Kerr Lake relation (hyphens, not underscores, per the BCP 47 standard). It feels weird to use ISO 3166-2 codes here, as though there are two dialects called “North Carolina English” and “Virginia English”, but it gets the point across. I also changed the name to Kerr Lake;Buggs Island Lake, accounting for the fact that it mostly lies in Virginia.
Regardless, we don’t split the lake along the state line, because it’s still one lake in reality, and both states apply the name to the entire lake. This “dialect”-based solution also applies in cases where a jurisdiction insists on a particular name for a feature that it does not claim for itself at all – probably relevant to seas and the like. But it doesn’t absolve us from having to pick one or more names among the various possibilities.