The No True Scotsman fallacy, also known as Appeal to Purity is a logical fallacy that seeks to define a thing in terms of what the person making the claim (or appeal) considers to be pure.
For example, a Hindu who believes India belongs only to Hindus might be confronted by another Hindu who does not think so. He would then seek to redefine "Hindu" as only Hindus who agree with him - "true" Hindus. This would involve him saying that Hindus who don't agree with him about India belonging only to Hindus are not "real Hindus" or "true Hindus".
Similarly, a Hindu who wishes that India should belong to people from every religion will call the narrow-minded person from his own religion to not be a "true Hindu" by trying to define Hinduism as a more open and inclusive faith.
In both cases, the same logical fallacy is being made. In reality, there is no "true Hindu". But people keep saying so because they need to continue thinking of themselves as Hindu. The bigot cannot lay claim to being Hindu if Hinduism is what the liberal Hindu is saying. The liberal Hindu cannot lay claim to being a Hindu if Hinduism is defined using the bigot's terms.
Neither will abandon the Hindu label though. And the reason behind that is mostly that the label "Hindu" is too useful to them. Perhaps it is the religion they have grown up with and it helps them identify themselves. Perhaps it is useful because of all the privileges it grants them - privileges they are unwilling to let go of.
At the end of the day, the blunt truth is that Hinduism is not the most important thing in the world. And when we have to choose between holding on to it for comfort and abandoning our need for it so we may be free to favour values like equality and empathy, we should not hesitate.