Up until 1889 and the creation of the County of London, "London", in the strictest sense refers only to the City of London (the boundaries of which have barely changed - this is the tiny little blob that is basically the areas surrounded by the old Roman walls). The areas that we now consider today as part of inner London belonged to the four surrounding counties - Kent in the Southeast, Surrey in the South and Southwest, Essex in the East, and Middlesex to the North and West. There was no unified government responsibile for the entire conurbation, just individual governments for the various constituent boroughs.

For practical purposes, there was a need to provide the same services to them all through the same organisation. Using "Metropolitan" as an adjective was therefore a convenient way of designating that the body was responsible for the area around London (as opposed to saying Northwest Kent-Northeast Surrey-East Middlesex Board of Works), and many of these Metropolitan institutions were mergers of various previous local boards, because London itself had a distinct legal meaning (and indeed, the City of London has its own, seperate police force, called the City of London Police, not part of the Metropolitan Police!)

Metropolis was never really used as a substitute for "London" as a place name and geographical location beyond these administrative contexts (especially as "Metropolis" is in itself a generic adjective) - the various British Railways that made their home in the London area all named themselves after London - e.g. London and North Western Railway, London and South Western, London, Brighton and South Coast, London, Chatham and Dover. All these were founded in Victorian times in the decades around the formation of the Metropolitan Police/Board of Works. Almost none of them actually had actual stations in what would legally be considered the City of London (other than the London, Chatham and Dover's station at Blackfriars), but all used London in their name to highlight the fact that they do, in fact, go near London.

As for London vs Westminster, the conurbation was always referred to as London, and Westminster (city status 1540) grew up as an outcropping of London (city status time immemorial); You can see from this map held by the British library (http://www.bl.uk/onlinegallery/onlineex/crace/m/largeimage87965.html) that the density of the conurbation is centred around what is now the city of London; the bits near Westminster are clearly a lot less dense and, despite its city status, it was clearly not the focus of the city.