How the word "Hinduism" came into picture?

  

" There is a word which has become very common as an appellation of our race and our religion. The word “Hindu” requires a little explanation in connection with what I mean by Vedantism. This word “Hindu” was the name that the ancient Persians used to apply to the river Sindhu. Whenever in Sanskrit there is an “s”, in ancient Persian it changes into “h”, so that “Sindhu” became “Hindu”; and you are all aware how the Greeks found it hard to pronounce “h” and dropped it altogether,so that we became known as Indians. Now this word “Hindu” as applied to the inhabitants of the other side of the Indus, whatever might have been its meaning in ancient times has lost all its force in modern times; for all the people that live on this side of the Indus no longer belong to one religion. There are the Hindus proper, the Mohammedans, the Parsees, the Christians, the Buddhists, and Jains. The word “Hindu” in its literal sense ought to include all these; but as signifying the religion, it would not be proper to call all these Hindus ." - Swami Vivekananda

(Complete Works of Swami Vivekananda . Swamiji's speech at Madras)

How the word "Hinduism" came into picture?. Here swamiji is explaining the origins of the word "Hinduism" and  can it be used in the present day context. In the later part of the same lecture he explains what's the proper way of calling the followers of the sanatana dharma.