I live in TW.
I pay council tax to LBRUT- London borough of Richmond.
I ca walk 20 mins and be on a London tube.
In a drop down list on most websites when I'm inputting my address, it's 50/50 whether my option is Middlesex or London/ greater London.
If anyone where I'm from, Cornwall, asks where I live, I say either twickenham or Richmond (depending how sporty they are, if they know where I mean!) in greater London. Richmond, or surrey, or Middlesex, means bugger all to most people I know so London clears it up.
It's a grey area, but I've never had anyone get riled about whether I say London or Middlesex! I'm literally right at the edge of Middx, few streets over is surrey, does anyone care?
Council tax goes to a London borough. Hence London.