When I visited London in the late '70's, most of the street signs said "City of Westminster". I was told that the "City of London", properly speaking, was only a small part of what outsiders would call "London", and that the Lord Mayor of London was only the Lord Mayor of that much smaller part.

From what folks are saying here, I gather that that's changed; but that's how it was then.

Aha: Wikipedia on "Greater London": https://en.wikipedia.org/wiki/Greater_London