yes Savarkar was an atheist, he took to cultural nationalism cause he found it to be the only thing which can be used to defend the people of the nation against Islamic aggression.
"Bharat" is official name for India in Hindi/Sanskrit adopted by the constitution. Hindustan is also a term used to define India. The term was used by Muslims rulers who invaded India(500 years back) calling it Hindustan (Place where Hindus are found).