Indeed. The word Hindu does not occur in native literature, its an exonym https://en.wikipedia.org/wiki/Endonym_and_exonym

If I remember correctly earliest recorded use of the word is in Persian/Zoroastrian text. There it's used not as a name of a religion but as a name for people living on the other side of the Indus river (called Sindhu in Sanskrit)

Correct. The right name is 'sanaatana dharma'

https://en.wikipedia.org/wiki/San%C4%81tan%C4%AB

Calling it the right name is excessive. The term is of fairly recent origin and motivated by a desire to avoid using a Persian loanword.