I was editing an article that referred to the Wörthersee, in Austria. As it stood, the copy referred to “Lake Wörthersee,” but a reader who speaks a little German might know that See is German for “lake,” and find that wording redundant.
A fellow editor who used to work for a news service in Japan told me that their house style was to replace the Japanese suffix for the geographical entity with the English word: Fujisan would become “Mount Fuji.”
The Chicago Manual of Style recommends leaving out the English term: “the Rio Grande” not “the Rio Grande River.” (Chicago Manual of Style, 16th ed., 8.54 “Foreign terms for geographic entities.”)
For the Wörthersee, I chose “Lake Wörther” instead of “the Wörthersee,” since a) it isn’t a very well-known place, so I didn’t think changing the name would cause confusion; and b) the German for “lake” isn’t that widely known. Based on the same considerations, I would leave Rio Grande, which is more widely known to North American readers, as is instead of changing it to “the Grande River.”