We have two correct names. Both the names are written in our Constitution - Bharat & India. Feel free to use either one. You as a citizen of Pakistan are not entitled to decide what is the correct name and what is not. India is also a historical name though one that is less ancient than Bharat.
I wonder why you say people get embarassed about Bharat. You must have heard us saying Bharat mata ki jai.
Actually, it is the Muslims who were ordered by your founder to kill innocent people on Direct Action day to press his demands. And the Muslims started killing and pillaging.
Even despite that, during partition itself Muslim mobs started massacring Hindus and Sikhs wherever they could find them in small numbers, women and even children were raped. Which is why today Pakistanis still have a culture of bacha bazi as an example.
Pakistan and Pakistanis have merely continued the culture that its founder espoused - that of mass murder, violence and persecution.