India the name comes from the latin greek word Indus.
I will tell you the real picture, of language, which made so many changes in the name of our country.
Long centuries ago, persian and Arabs came to india as for trade purpose. Before that India was called as Aryavarta, Mean land of Aryans(nobles).
Persians found a major river having 7 tributories of it. People at that time called it Sapth Sindhu. As Sindhu river having 7 adjoining tributories.
Persians pronounced /s/ as /h/. So sooner it became HAPTH HINDU.
In short they called it HIND. Greeks taken it as Indu(s). And modern latin word named it as INDIA.
When Mughal came to India, they called it as HINDUSTAN(HINDU+STAN). STAN in mughals means “place to live”.
The name Bharat was given long time back as of on the name of King Bharat.
He was the first person to conquer all world and united it into a single name as Bharatvarsha.
He was the son the King Dushyanta and Shakuntala. He was brave since his childhood as he counted teeths of a lion.