We present a collated set of interaction cross sections for positrons in water, and study how the choice of ionisation energy sharing and of anisotropy in elastic cross sections influences low-energy spatial transport. A Monte Carlo code has been developed to model the transport of a beam of positrons injected at 60 eV into water, in which we compare several ionisation energy sharing models, and also vary the anisotropic scattering behaviour for elastic collisions. While the cross sections are primarily applicable to water vapour, we have investigated the inclusion of coherent elastic scattering which is present in liquid water. We present comparative profiles of positron number density, energy deposition, positronium formation, and secondary electron generation. Anisotropic scattering can increase radial diffusion by a factor of two, and a similar effect occurs as a result of coherent elastic scattering, though only at energies below the positronium formation threshold. The results emphasise the need for detailed knowledge of scattering cross sections that are differential in both scattering angles and energy transfer.