Actually there are accepted technical terms for both senses of distance, and therefore 'rest', in cosmology. One refers to the comoving distance versus the Fermi distance. Either one is a proper distance in the sense of being a geodesic distance. One is computed using hypersurfaces of constant cosmological time, the other using a Fermi-normal slice based on a particular comoving observer. Unfortunately, in cosmology, it is common to allow unqualified proper distance to be the comoving distance.But I think a lay persons who understands the meaning of "the proper distance between the objects stays constant" refers this to flat Minkowski spacetime and not to the expanding universe. Regarding the latter he refers "at rest" relative to the CMB or relative to the isotropic universe. A lay person who doesn't have this very basic knowledge doesn't understand the meaning of "the proper distance between the objects stays constant" and doesn't even think about the meaning of "at rest".