There are an overwhelming number of mathematical models which are referred to as Boussinesq equations. This may easily lead to confusion, since often they are loosely referenced to as the Boussinesq equations, while in fact a variant thereof is considered. So it is more appropriate to call them Boussinesq-type equations. Strictly speaking, the Boussinesq equations is the above-mentioned set B, since it is used in the analysis in the remainder of his 1872 paper.
Some directions, into which the Boussinesq equations have been extended, are:
*varying bathymetry,
*improved frequency dispersion,
*improved non-linear behavior,
*making a Taylor expansion around different vertical elevations,
*dividing the fluid domain in layers, and applying the Boussinesq approximation in each layer separately, 
*inclusion of wave breaking,
*inclusion of surface tension,
*extension to internal waves on an interface between fluid domains of different mass density, 
*derivation from a variational principle.