Frank Fellone's column in today's Democrat-Gazette explains why the D-G's executive editor Griffin Smith has decreed that Mumbai shall remain Bombay in the pages of that newspaper, but not most other U.S. news organizations. It's no different than the U.S. custom of referring to the Italian city of Firenze as Florence. Only English spoken here, see.
I'd argue the comparison is not strictly parallel. The Indian language situation is far too complicated to summarize, but English is one of the country's official languages. English-speaking Indians call the city Mumbai, too. Not that what they think or say should matter to a D-G editor, of course.
Newspapers are quirky things. For years, an editor decreed that the Arkansas Gazette would refer to the Metrocentre Mall (remember that?) as the Metrocenter Mall, on account of the editor's dislike of the fancified spelling. That was a stupid call, too, much derided by newsroom peons (as the colonialist hold on Bombay is derided by at least some at the D-G). WWDODD?