Water is a greenhouse gas, it's true, and on a per molecule basis has a similar magnitude of infrared absorbance - and therefore radiative forcing potential - as CO2. However, the effect of water vapor on global atmospheric heat trapping is strongly dependant on its average atmospheric concentration. That, in turn, tends to be strongly heterogeneous regionally, varying on Earth from about 0 to 4% depending on latitude, average surface temperature, wind patterns, and availability of evaporating liquid water at the surface.

On a planet as described in the narrative, absolute global H2O vapor concentration would be likely limited by freeze-out on the night side and low availability of surface liquid on the day side. So the relative humidity, and therefore the absolute concentration of water vapor, would be globally low, with a relative maximum near the terminator (over a liquid ocean) and a minimum at both poles. So, the average global effect of H2O on absolute radiative forcing would likely be minimal.

As far as a higher volatile content producing a global ocean, that is one potential model, and it may be more accurate. However, the model described in the narrative seems to suggest that due to tidal locking effects, at any given moment much of the available H2O reservoir is held as a solid on the night side. This could be a realistic model, given that the rate of viscous relaxation of ice would be the rate-limiting step in delivery of liquid H2O to the terminator environment. That step would likely be slower than the rate of H2O vapor delivery to the night side, resulting in a low average global concentration of H2O vapor and a limit to the amount of liquid water there could be on the day side. It would probably take some rheological calculations of the rate of viscous relaxation of ice in the heavier gravity, along with the significant distance (Gliese 581c is probably a larger planet than Earth) the glaciers would have to travel to reach the terminator, and some estimate of the total reservoir of surface H2O, to make a definitive conclusion.

One thing the author didn't mention but which would probably be a factor on Gliese 581c; crustal strain produced by tidal effects. On a sphere experiencing anisotropic stress, you'd expect strain fractures along the equator perpendicular to the axis of stress (in this case the tidal pull direction, oriented to the day/night 'poles'). Along the terminator there would be latitudinal rifting of the crust, forming immense grabens into which a liquid ocean would tend to accumulate. The deepest lake on Earth is Lake Baikal (Russia), formed over a graben resulting from continental rifting. It's likely a similar process would occur on Gliese 581c, but there resulting in an equatorial ring of graben-bound seas.

With so many variables, it seems the best we can do without direct evidence is come up with a suite of possible models constrained by known factors. That still leaves room for some fun speculation, for now.