There are good historical reasons, yes, but the balance shifted about 200 years ago. For a long time Charing Cross was treated as a semi-official designation of the centre point of London, for good reason.
> In terms of modern London though, I take it as defined by the M25, and on that scale, the everything from Greenwich to White City is pretty much central.
If you take expansive definitions like that, sure. But if e.g. people tell you they're "going in to London", odds are they're talking about Westminster unless they happen to work in banking.
There are endless definitions of this, but the point I was making is that when people think of London, City is rarely what they're thinking of. You see it even in the article, where they chose to lead with a picture from the centre of Westminster and describes it as "in the heart of London".
EDIT: This is particularly relevant in context of your original comment about the "essential purpose" of London: Westminster was nerver part of Roman London/Londinium - it was formed as a separate town. It's purpose was entirely different. What you describe as central London includes a number of other old towns with entirely different purposes to City.