The traditional many-body expansion-in which a system's energy is expressed in terms of the energies of its constituent monomers, dimers, trimers, etc.-has recently been generalized to the case where the "monomers" (subsystems, or "fragments") overlap. Two such generalizations have been proposed, and here, we compare the two, both formally and numerically. We conclude that the two approaches are distinct, although in many cases they yield comparable and accurate results when truncated at the level of dimers. However, tests on fluoride-water clusters suggest that the approach that we have previously called the "generalized many-body expansion" (GMBE) [J. Chem. Phys.137, 064113 (2012)] is more robust, with respect to the choice of fragments, as compared to an alternative "many overlapping body expansion" [J. Chem. Theory Comput.8, 2669 (2012)]. A more detailed justification for the GMBE is also provided here.