The classical Latin was "India" taken from the Sanskrit "Sindhu" and Persian "Hindu" possibly referring to any large river which formed a natural border. The Greeks picked it up from the Persians as Indós (Ἰνδός) and Herodotus talks about India, some of it is actually true.
Eastward of India lies a tract which is entirely sand. Indeed, of all the inhabitants of Asia, concerning whom anything is known, the Indians dwell nearest to the east and the rising of the Sun.
Take Herdotus with a heavy pinch of άλας.
The Romans borrowed from the Greeks as Indus, India, and Indian. Old English borrows it from the Latin as "India" or "Indea".
The area referred to as "India" began with the Persians and Greeks referring to the area around the Indus River. Then everything east of the Indus River. To the Romans and Europeans it could refer to just about anything in South Asia west of China, or even the horn of Africa. Thus we get the "East Indies", the misnamed "West Indies", the "East India Company", and "Indonesia" (an 18th century melding of Ἰνδός (Indus) and νῆσος (Island)).
The country we call "India" is a product of geography and British colonialism. Prior to the British the region was made up of many nations and empires. In some periods nearly all of modern India was under the control of a single empire such as the Maurya around 300 BC, to the Mughal Empire established in the early 1500s until its decline in the 18th century. The Mughal Empire and the British East India Company were both dissolved and the Government of India Act 1858 established the British Raj in 1858. This was partitioned in 1947 into the Dominion of India and the Dominion of Pakistan. The Dominion of India became the modern Republic of India.
Myanmar / Burma
This is already well covered on Wikipedia by Names of Myanmar. Here's a tl;dr.
Burma comes from the Bamar or Burmese people (ဗမာ [Bamar] + လူမျိုး [race]). It's thought that colloquial Burmese shifted the B to an M with many different variations through Burmese history. မြန်မာပြည် or Mranma kingdom, spoken with a soft R, is "Myanmarpyi".
The change from Burma to Myanmar reflects attempts to shed their colonial legacy. "Burma" likely comes from the Indian "Barma" possibly a mixing of "Bama" and "Brahma-desh". The Portuguese turned this into Birmânia (turning "Birmans" into "the country of the Birmans" Latin-style like "Germania") which came into English as however 18th century map makers thought it should be spelled: Bermah, Birmah, Brama, Burmah, Burma, Burmah. The Times finally standardized on Burma.
Like India, Senegal refers to the region around the Senegal River named so by the Portuguese for... something, it's not clear. There's a number of hypotheses including the Zenaga language of the region or a misunderstanding of "Sunuu Gaal" meaning "our boat".
Explorer: "What's that?" (points at river)
Local: "Our boat." (thinks they're pointing at their boat)
Explorer: Ahh, the Ourboat River.
It's probably a folktale, similar to the idea that "Canada" is Cartier misunderstanding the local word for "the town" despite Cartier's own writing correctly translating the word.