Myanmar:

Myanmar is still called Burma. The two names are two different registers. Imagine it like saying "the States" versus "America". Not the best analogy, but the point is that the name didn't change. Actually as I think of it that's a horrible analogy. If you speak Chinese I can give you a way better one.

The former, Myanmar, is a more formal version of the name. Burma is less so, and also refers to the majority ethnic group, often rendered as Bama in English. Note the final -r on Myanmar isn't really there. The final syllable of both words is identical, even though it's not written that way in English. Maybe a better analogy is "father" vs "papa"; They mean the same thing, one's just a more formal version.

Anyway, within Myanmar, both are used. It's not that the name of the country changed. It's that the official English name changed. But in Burmese we're still saying both.

The general issue for people here is actually that ဗမာ (/bəma/, "Burma") is also the name of a major ethnic group without the same register distinction in play to the extent that it is with the name of the state, so မြန်မာ (/mjəma/ "Myanmar") is intentionally more inclusive, as the narrative goes.

However both names are actively still used in Myanmar today, by ethnic Burmese and by Kachin, Naga, Shan etc, even in areas with strong ongoing separatist movements.

The full formal name of the country is the Republic of the Union of Myanmar (ပြည်ထောင်စု သမ္မတ မြန်မာနိုင်ငံတော်‌). The old name was the Union of Myanmar (often the Union of Burma in English, but in Burmese is ပြည်ထောင်စု မြန်မာနိုင်ငံတော်‌, note the "မြန်မာ" part which is Myanmar).

Thailand:

"Siam" is more interesting, if only because it's precise etymology is unclear. In Thai, Thailand has never not been called Thai[land]. The Dai ethnicity in China are the same ethnicity, and are also called by the same name (what's written D here is actually /t/). The Shan, ethnic Tai's in Myanmar, call themselves Tai. The Ahoms in India are also called Tais among themselves.

This gets more interesting in those areas because the word "Ahom" (/oxom/) is also where the word "Assam" (the state in NE India) gets its name. The /s/ became /x/ (written <h>) but in many of the semi-indigenous languages of the region, the Ahom's are still called "Sam". It's the same original word just branched out in different ways for different areas. Just across the border, the sister group, the Tai Shan's, are still called Shan by the country and by most people, but again don't call themselves that. Their own name for themselves is Tai. So it's not just Thailand where you see this Tai/Sam dichotomy. It's present among most Tais throughout SE Asia.

Back to the etymology. The word Siam most likely comes from a Sanskrit word. It was an exonym applied to Thais by non-Thais. It was picked up as the English name of Thailand in the 1800s, but then was dropped for a brief period. The official English name became Thailand again after the establishment of a constitutional monarchy.

To Thais though, it's always been Thai[land].

tl;dr:

The name of the country only changed among outsiders. Within those countries the core of the name has been constant.

If you want sources I can probably dig them up but much of this is bordering on common knowledge for linguists working in the area. If instead you'd like further reading, that's a bit easier. For example see Myanmar: State, Society and Ethnicity about the Shan.