A rigorously correct and fully quantum-mechanical theory of slow atomic collisions is presented, which removes the formal defects and spurious nonadiabatic couplings of perturbed-stationary-states theory, and arrives at coupled equations for the heavy-particle motion which are the same as those obtained in the preceding paper by the electron translation factor formulation. Here, however, the theory is formulated in terms of suitably defined scattering coordinates, and electron translation factors do not appear. A unified physical interpretation of both approaches can thereby be made, and smaller terms in the coupled equations, describing corrections of order mμ to electronic binding energies and to the collision kinetic energy, are placed on a firmer footing. Particular attention is paid to the critical test case of isotopic systems such as HD+ and it is shown how a correct theory of isotopic charge exchange can be formulated.