Sindhu indeed means river, and by extension a large body of water. The etymology is not clear, but maybe indicates the flowing nature of a river from a root sidh meaning to go. The word Sapta means seven as could be guessed from the cognate sept- (Of course the Indians never knew how to count: roughly the same bunch of rivers gave Punjab its name. panch is cognate with penta- and apas are the waters :-) More seriously, rivers change courses, so their numbers also change.)
I am not sure if you meant to suggest that the old persians called Sindhu as India: as you correctly point out later in your post, they changed the S to an H.
I was nicknamed Indeo by someone in the US once. Never heard of it as a formal name.