Ah I see, thanks for clarifying.The expansion scalar for the co-moving congruence is invariant - any coordinates at all can be used to compute it. What Peter was noting was that in standard cosmological coordinates, it is the same as the coordinate dependent expression he gave.
This is a good example. In contrast to proper distance the expansion scalar describes true physics (in the sense of being not observer dependent), the breaking of the string.Relating this back to the example we were discussing of rockets connected by string with the rockets having a specific thrust profile, the inertial frame would compute that proper distance between the rockets was decreasing, but the string would still break. If the expansion scalar for the string congruence were computed in inertial coordinates it would show expansion of the string, despite the measured decrease of proper distance between the rockets over time.