I revisit the problem of the interaction between two dissimilar atoms with one atom in an excited state, recently addressed by Berman [Phys. Rev. A 91, 042127 (2015)], Donaire et al. [Phys. Rev. Lett. 115, 033201 (2015)], and Milonni and Rafsanjani [Phys. Rev. A 92, 062711 (2015)], for which precedent approaches have given conflicting results. In the first place, I discuss to what extent these works provide equivalent results. I show that the phase-shift rate of the two-atom wave function computed by Berman, the van der Waals potential of the excited atom by Donaire et al., and the level shift of the excited atom by Milonni and Rafsanjani possess equivalent expressions in the quasistationary approximation. In addition, I show that the level shift of the ground-state atom computed by Milonni and Rafsanjani is equivalent to its van der Waals potential. A diagrammatic representation of all those quantities is provided. The equivalences among them are, however, not generic. In particular, it is found that for the case of the interaction between two identical atoms excited, the phase-shift rate and the van der Waals potentials differ. Concerning the conflicting results of previous approaches in regards to the spatial oscillation of the interactions, I conclude, in agreement with Berman and with Milonni and Rafsanjani, that they refer to different physical quantities. The impacts of free-space dissipation and finite excitation rates on the dynamics of the potentials are analyzed. In contrast with Milonni and Rafsanjani, the oscillatory versus monotonic spatial forms of the potentials of each atom are found not to be related to the reversible versus irreversible nature of the excitation transfer involved.