Is Kingston South London?

Royal since time immemorial (pre 1189), home of Coronation Stone, Clattern Bridge (ca. 1175), and Bentalls (1867), Nipper's (1884-1895) resting place, launchpad for Cher's Believe (1998) comeback sure, but South London? Is that not Lambeth, Southwark, or Wandsworth more like? It was referred to as Surrey when I first moved here to study 25 plus years ago, but even then that seems to have been incorrect. The geographically more precise South West was short lived as formal designation, so I dug up a few key dates how the remapping came to be:

1441
Incorporation, royal manor up to that point
1481
Given borough charter by Edward IV
1836
Made municipal borough under the Municipal Corporations Act 1835 (5 & 6 Will. 4. c. 76)
Name definitely includes hyphens
1927
Royal borough status confirmed after petition
1965
No longer Surrey, made London borough 23 under the Greater London Act 1963 (c. 33)
Royal status confirmed by Elizabeth II
Name loses hyphens
2004
Assigned to South London (London Plan, ch. 5F)
2008
Changed to South West London (London Plan, ch. 5E)
2011
East and West sub-regions merged back into one of South London (London Plan, ch. 2.5)

Until 2020, interestingly enough, Kingston was the seat of Surrey County Council, despite being no longer administered by it.