Here is some history from Matthew Paul:
From Report on Richmond upon Thames’s mixed community secondary schools’ Linked Schools Policy: September 2011:
1990-1991: . . 2.1 The admissions policy for the borough’s community secondary schools for 1990 admissions gave priority to children resident in Richmond upon Thames . .
2.2 By 1990, though, local authorities had absorbed the full ramifications of the High Court Judgement against the London Borough of Greenwich in 1989 (which) established that it was unlawful for a local authority to give priority in a school’s oversubscription criteria on the basis of a child’s residence within the authority’s administrative area. In other words, whilst it was – and remains – lawful to use distance from home to school as a criterion, applicants from outside the authority’s area should otherwise be considered for admission no less favourably than children living within the area.
2.3 Following the judgement, the authority experienced a considerable increase in the number of applications for its secondary schools from out-borough residents. It was feared that the schools’ popularity, the elongated shape of the borough and the schools’ relative proximity to the borough’s boundaries would lead to an excessively large number of children from outside the borough obtaining places in the borough’s schools. This would upset the traditional patterns of transfer between primary schools (both within and outside the borough) and the borough’s community secondary schools, and could lead to a shortage of places for in-borough residents.
The authority therefore established the Linked School Policy (LSP) in order to encourage a fair degree of stability in its admission patterns. The LSP also ensured that a large number of Year 6 pupils would reap the benefits of educational continuity by transferring with their peers. The authority believed that the community at large would be made more cohesive as a result.
2.4 The LSP for 1991 designated certain junior and primary schools as being ‘linked schools’ (incuding some out of borough schools) . . Waldegrave School for Girls was linked to all of (them) . .
2.10 The ‘quadrant system’ – four geographical areas covering all of Richmond Borough and part of the five neighbouring local authorities, to which places were allocated according to demand from each area – was introduced for admission to Waldegrave School for Girls in 1992. However, priority was still given to “girls attending a link primary school” . .
2.16 For 1997 entry, an ad hoc sub-committee of the authority’s Education Committee recommended that the criterion regarding the LSP should be removed from the oversubscription criteria for Waldegrave . .
So the current 'A' and 'B' areas are a simplification of the 4 quadrants adopted 11 years ago. I do not know when this change came in.