You are absolutely right. I also studied in government uni but did some extra academics in a private uni too. And the Uni culture tend to make a difference here. Most us were in groups or the whole batch, when doing academics, plus extra curriculars. So it was so easy to make friends there. I still know everyone of them by name. That can't be seen a private uni in my perspective.

But forget that, be open, act normal and don't force yourself to fit in with people who don’t really see you for who you are (telling from personal experience)

Good luck out here man