Unfortunately, nobody took into account the difficulty of talking about this to people online when the County of London was created (totally excluding the City of London) in 1889, nor when the City of London and the London County Council boroughs were finally merged into Greater London 1965 (only to be separated again in 1986 when the Greater London Council was abolished, before being merged again in 2000 when the current Greater London Authority was created).

[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.