Martin Underwood writes
The boundary between the cities of Westminster and London [1] is very
short, probably only 500m or so in length - but even there the
boundaries are not precisely the same.

See http://tinyurl.com/im8l

[1] In their modern administrative boundaries.

There's a red boundary line that runs from near Chancery Lane tube
station roughly southwards to the middle of the Thames near HQS
Wellington. Is that the city boundary or the postcode boundary?

I assumed it was the postcode boundary, on the map the postal districts
are written in a font of the same colour & weight.

The thick greyish line would appear to be the borough boundary line.

I've never understood the rules governing postcode boundaries.

Simple. They are drawn purely for the convenience of the Royal Mail, to
enable them to deliver mail effectively.

--
Dave