India's main official name is India. Bharat (from Sanskrit) is coequal according to the constitution, and Hindustan (from Persian) is also sometimes used.
That's mostly because English is one of the two official languages of India, and even the local native languages (at least as spoken by the English-educated middle class) are heavily influenced by English. Similar to how the Philippines are just called the Philippines (or Pilipinas, the closest approximation that is pronounceable in Tagalog), rather than a native-language name.
