The City of London and the city of London are distinct legal entities.
It is most emphatically NOT "London, the city", in exactly the same way that Civic center is not the city of San Francisco.
It's just confusingly named.
Parliament, for example, is in the City of Westminster, one of 32 boroughs which (+ the City) make up Greater London.
"the city of london" is just a confusingly named area inside a city called "London", it's not a city.
Normal people where?
London has only had a mayor since 2000. Each borough has its own council, there is no London council.
You don't know what you're talking about and you're projecting (North) American ideas onto other countries.
> so when people talk about a city called London it's safe to assume they mean the 32 boroughs.
Except it's not. Colloquially, some borough are not considered to be in London. Bromley, for example.