I live in a London borough, have an 0203 (the new dialling codes) phone number (everyone else here as 0208), use the local tube station and buses (with my Oyster card), but my postal address has MIDDLESEX as the county.
But- this is where it gets confusing: 
We moved here from a W9 postcode- (I am a proud Londoner)
When we go on holiday we tell people we live in West London, because technically I do live in West London, but not the West London that I used to live in! (And I am too ashamed to admit where I live for fear of being judged or stereotyped)
If we go shopping we will go 'into London' and we sometimes take the mickey out of the people who live around here because they are from 'the country and you can tell they are not from real London' ( I may sound like a snob but you would too if you had to live here)!
But, I would never admit this and maintain to my friends that yes, of course I still live in London. I am also bringing my daughter up to believe that she lives in London (as I don't want her to identify with the great unwashed that live around here ps. we do intend to move).
BUT BUT BUT technically we DO live in London, and I cling on to that...
ALTHOUGH- if I had never moved and still lived in real London, I would be on the OP's side and not class anything out of W,WC,NW,N,E,SE,SW postcodes as London.
HTH 