When the military government changed the country's English name from Burma to Myanmar in 1989, a lot of Burmese people overseas simply kept calling it Burma. It was not a political stunt on their part, most of the time. It was the name they grew up with, the one their parents wrote on envelopes, the one that came out naturally in an English sentence. A name you hear from a young age has a long shelf life.
Burma and Myanmar are both derived from the same root, the Bamar people. Neither is more "authentic" than the other in any strict sense. The change was about official English usage, not about the Burmese language, where the country has had the same name all along. So the real question is not which one is correct. It is which one a person means when they have settled on one. Usually that tells you more about the speaker than about the place.
The same thing happened to the cities. Rangoon became Yangon on paper, but you still hear Rangoon everywhere, from people in their sixties mostly, and from the news in English. It is the same town. The streets are the same streets, crowded, loud, green at the edges, and the heat does not care what you call the place.
Within a single family you will hear both. A cousin who left recently says Yangon without thinking. An uncle who has been abroad for thirty years says Rangoon. They are not disagreeing about anything. They are just dating themselves, gently, with the name they happened to be handed.
Then there are the older spellings. Pagan becomes Bagan in the romanised forms used now, and the temples in the plain are just as beautiful under either spelling. You will still see Pagan in older books and in poems. Nobody has a strong opinion about it the way they do about Burma and Myanmar, because it was a colonial-era English spelling, not a recent official one, and it never got tangled up in politics the same way.
Around the dinner table, nobody uses either English name much. We say the country by its Burmese name, and we say it quickly, the way you say the name of your own house. The English words Burma and Myanmar are for talking to people who are not from there. When we switch to English, most of us say Burma without deciding to, out of habit and warmth, even some of the people who would not mind Myanmar at all.
So if you are trying to get it right as an outsider, the honest advice is: Burma is the safer, warmer, more commonly used choice among the diaspora, and nobody reasonable will be offended by Myanmar either, as long as you are not using it to take a side in a fight you were never part of.
Names matter here because they carry memory, and memory is what a community abroad spends a lot of energy holding on to. The country many Burmese people miss is not exactly the one on a current map. It is the place under the old name, the one that existed before, in their heads. Noticing that a person chooses Rangoon or Burma or Yangon with care is a small way of listening to them, and listening is most of what this site tries to do.
If you want the longer story of how the country itself changed, not just what it is called, our timeline of Myanmar's history is the place to start.