I don't think there's any harm with using the term "proper distance", especially since the Omega parameters in both astcosmiccal and Hogg 2000 don't bother distinguishing their current values from their general values (showing how matter density changes proportionally to (1+z)^3 due to expansion would be written in that terminology as Omega_M = Omega_M (1+z)^3 for arbitrary z, literally implying that z = 0, defeating the point of having the equation). It will generally be understood that "proper distance" means "proper distance on the current time slice according to the FLRW foliation". Hogg's point is that "proper distance" can, in principle, be done on a time slice at any epoch. But his language is not really careful enough anyway, since "distance between them which would be measured with rulers at the time they are being observed" doesn't make sense for observing a single distant galaxy: we are at t_0 and the distant galaxy is at t << t_0, so there is a mix of "times" at which the distant galaxy is observed. If we project small intervals to each of the time slices between us and the galaxy, we get the lookback time in units of distance, not the radial comoving distance.

All the same, changing from "proper distance" to "comoving radial distance" would be reasonable, since it would remove any ambiguity; "comoving line-of-sight distance" would be OK too, since it's unambiguous.

The "comoving distance (tranverse)" makes more sense for "proper distance", because that occurs at a single redshift/epoch - but in general people don't say the "transverse proper distance". And in any case, that's not calculated by astcosmiccal .