If we want to shorten the name of City of London, the short form is the City, not London. The only people who ever calls City by the name of London is people who have just learnt of the oddity.
Greater London makes up the formal boundaries of London. It's the legal and administrative boundary, and the city limits of London. Unusually, unlike the city limits of many other cities, you'll find plenty of people including people who live here who not quite consider the outer parts to be part of London. People where I live sometimes still consider it part of Surrey even though it's been part of London since 1965.
There are forests and agricultural areas almost entirely separating parts of the borough I live in from the rest of London.
Yet in other areas, the Greater London urban area expands well past what anyone would call London, including e.g. entirely separate towns in other counties, like Watford in Hertfordshire, Gravesend in Kent, Epsom and Guildford in Surrey.
So yes, these comparisons are tricky but that is the point.
E.g culturally, London is likely reasonably described as smaller than it's city limits, while indeed some other cities are often larger than them, but rarely as large as the urban area they are within. It's extremely common for towns on the outskirts of large cities to have their own separate identities and not be considered part of the adjoining city, but still form part of the same urban area.
Depending on what you want to compare and which cities you are comparing, different measures will be more or less appropriate, and may require different considerations.
This seems almost obnoxiously confusing when talking online to people from all over the world.
[And, yes, this means there was a 14 year period when the UK had a capital without a government, nor indeed any administrative unit - effectively "(Greater) London" as a political and administrative entity didn't exist during that period -, largely for political reasons - Thatchers government was strongly at odds with the then particularly left wing Greater London Council]
But really, the reason for contracting City of London to "City" rather than London is to minimise confusion, because it means London fairly unambiguously, though not entirely, refers to Greater London.
You can have these differences within one city as well, on the district level or even individual streets.
For this discussion (cultural centers of Europe), Greater Paris, Greater London and Berlin (they have roughly similar areas) are what you want to compare, and Berlin is clearly the smallest one population-wise out of these three (and it's not even close).