Goodness, don't people feel strongly got about this! Surely, when most of us say where we live, we try to give the answer that is meaningful to the person who has asked (assuming we're not deliberately trying to hide something).
Over the years have lived in 2 areas that were indisputably London (NW and SE postcodes) and two areas that were in London boroughs, with London phone numbers, but which didn't have London postcodes (HA and TW).
If people I met in London asked where I lived, I'd have said eg Lewisham or Harrow. But if I was asked by someone from outside London, I'd probably have said I lived in London, and then would have been more specific if asked.
I now live in rural Surrey. If asked where I lived, I might say Surrey, because most people who aren't local will never have heard of the town I live in. But if asked by someone a few miles away, I'd say the town. If asked by someone local, I'd tell them the name of the road!
As to discussions about 'proper' London, obviously London is getting bigger all the time! I was born and brought up in the West Country, so I never considered myself a Londoner, even when I lived there. My grandmother, however, always claimed to be a proper cockney because she was born within earshot of Bow Bells.