Every so often, one hears debate about what is "the right name" for some country, city, language, or ethnicity.
Sometimes this happens when the rulers of a country rename it (with or without popular acclaim in that country), as with Burma becoming Myanmar. Sometimes this happens when there's a move to adjust the English name, whether as to the root or just as to the spelling or pronunciation, to be closer to the name in the local language.
Sometimes it happens when people simply object to the supposed "mispronunciation" of a place name that's derived from a foreign language, as a friend of mine did with regard to "Los Angeles" (the claim was that it should be pronounced "Los Anheles," much as "La Jolla" is pronounced "La Hoya" and not "La Dzhol-la"). And sometimes the argument takes a tone of grievance, for instance when the claim is that the English names for Canadian Indian languages are wrong because they improperly depart from the languages' own names. (Prof. Bill Poser [Language Log] has an interesting article on this latter question.)
What puzzles me is that these claims that certain English names are "wrong" seem inconsistent with the reality that geographical and ethnic designations are translated from language to language much as other words are. Deutschland is Germany in English, and Allemagne in French; the German language is also Nemetskiy in Russian and, I'm told, "lingua tedesca" in Italian. China is China in English and Kitay (cognate to the archaic English term Cathay), though to my knowledge neither of these is related to the Chinese term for China — both rather stem from the names of particular dynasties that have ruled China.
And of course even when the roots remain the same, even the spellings of names and most certainly the pronunciations change from language to language. Roma becomes Rome and London becomes Londres. Magyarorszag, Shqiperia, and Ellas become — well, I leave that as exercises for the reader, but I should note that they're in Europe, so it's not just dead white males oppressing the place names of color. People who complain about Los Angeles don't start calling Detroit Detrwah, though that would have been the pronunciation used by those who named the Detroit River, from which the city name flows. This is the way languages operate, and to say that it's "wrong" to call Beijing "Peking," to call Myanmar "Burma," or to call Los Angeles "Los Angeles" reflects a meaning of "wrong" that is at the very least not a linguistic meaning.
What can we say, then, about what geographical and ethnic terms (including the terms for the languages) are "correct" in English and which aren't?
1. As a matter of linguistics, those English names that are broadly in use, and that are generally understood, are correct terms for the thing named, just as the word "cat" is the correct term for a pet feline. At some point, the terms may become archaic enough that they stop being correct, in that either many people stop understanding them or they come across as so archaic that they interfere with communication more than advancing it. Cathay and Abyssinia are two examples.
2. Various people may for various reasons, whether political or esthetic, wish to insist that we change what we call certain things. Their desire to have us change how we speak does not itself impose on us any obligation, whether as a matter of morals or manners. Russians may not rightly insist that the English say Moskva instead of Moscow (though they are of course free to ask, if they so wish).
3. More broadly, there is nothing inherently demeaning or insulting in not using the foreign name for a foreign country, city, language, or ethnicity.
4. Sometimes, some new names refer to somewhat different locations than the original ones; then, it is misleading and, I'd say, incorrect to use the old name to refer to the new entity. The Czech Republic shouldn't be called Czechoslovakia — but because it's only part of the original Czechoslovakia, not because somehow the Czechs are just entitled to insist that we use the term they prefer. (There may also have been a similar Cathay/China distinction in English, though in Russian Kitay refers to all of modern China.)
5. For political reasons — whether grand politics or small politics — some people may choose to go along with some new name. That may make a lot of sense in certain contexts, for instance in diplomacy or business; but it doesn't create an obligation (moral, manners, or just general "correctness") on others to follow suit, at least until the old name becomes sufficiently unfamiliar or archaic (see item 1).
6. A rule that a name change becomes obligatory on English speakers would impose pretty substantial costs on people, as well as being inconsistent with past practice (not just in English but in other languages). First, telling people that they have to change how they speak is generally something of an imposition on the speakers.
Second, people will often continue to use the term they learned when they were young, just out of habit. If the use of the new word is seen as obligatory, such unintentional uses of the old word become seen as insulting (because they violate one's supposed obligation to the subjects of the word). The consequence is that the attempt to diminish supposed offense to the subjects will end up increasing actual offense, since what would otherwise be seen as an alternate name begins to be labeled offensive (even when it's not intentionally used this way).
Third, some proposed name changes create more confusion than they would resolve. "Native American" replaces one ambiguous term ("American Indian") with another, and in a context where the ambiguity may actually be more commonly sensed. Likewise, as Prof. Poser points out, many Canadian Indian tribes refer to themselves as some variety of "Dene," which means "people"; adopting those names would be confusing.
7. In certain situations, a particular term is so often used as a deliberate pejorative that subjects of the term reasonably assume that most uses of the term are pejorative; "nigger," "Nip," "Yid," and the like are classic examples. These terms do become improper, but only because they lead a reasonable listener to assume that some offense is intended. This is very rarely so at the level of standard geographical and ethnic names, as opposed to nicknames; I mention it here for the sake of completeness.
8. In the meantime, which name is right? Both the old and the new, so long as the new and the old are both familiar enough to be understood, and the old isn't so rare as to be archaic. People who use the old name should tolerate people who use the name without making a fuss about it; likewise, people who use the new name should tolerate those who use the old.
And when someone faults you for using the "wrong" name, where all they mean is the name that's different from that in the original language or different from what the country uses as its own name, well, you'll always have Paris — and Londres, Moscow, China, and Greece.