The name India was originaly introduced 2500 years ago by the Persians, who conquered the area of modern day Pakistan during the Achaemenid dynasty. Achaemenids named the whole area after the Indus River.
From Persians the name came to the Greeks and from Greeks to the Romans and from Romans to the peoples of Europe.
Indians themselves never called that area "India".
When British India was divided into India and Pakistan....both countries were thinking of how to name themselves after they gain independence.
Pakistan at first wanted to call itself "India" even ahead of India itself, but Founding Fathers of Pakistan thought that name "India" was associated with the shame of colonialism and they chose the name "Pakistan" instead.
Indians wanted to name their country "Bharat", but decided to retain the name "India" together with "Bharat".