If you look at the two screenshots, these are successive in the process of adding water vapour to Proxima Centauri II. You can see the Hydrographic Extent increases from 5.51 to 6.56.
Min Max thermal range changes from (-11.136 <> 54.498, which is a range of 65.634) to (-9.256 <> 56.849 range 66.105). So the range has increased not decreased and the temperature factor has gone from zero to 0.123 though it fluctuates quite a bit.
You need to let all that water vapor condense down to equilibrium, which will take ages from 0.5 atm. The increased range you're seeing is simply because you increased atmospheric pressure with water vapor which increased the body's greenhouse effect.
At the same time Hydrographic extent increased by 20% 5 to 6 ish but didnt seem to help and to the best of my knowledge does not have an effect, which is why I am suggesting liquid water could have an effect too.
I completely understand temperature increasing temporarily due to water vapour makes sense and the range of extremes increasing due to global warming also make sense as it is current meteorological science dogma and is considered to be due to the increased level of the energy equilibrium in the atmosphere causing higher wind speeds and thereby increased mixing of equatorial and polar weather systems in the atmosphere causing particular locations on the surface to experience a greater range of extremes.
But the global warming model does not include adding water to the system which would buffer the whole system as well as cause sea levels to rise of course.
In the example you could add Frigusium to counter the water vapour warming but that would only lower temperatures and shrink the range a little as a result, there is no way to change range independently of temperature so you are stuck with a fixed scale of range per temperature per body, as I understand it. On Proxima there is no way to bring the range inside tolerable though you might get away with straddling it.
If you are adding liquid water to a body then in theory the temperature range should decrease due to the buffering effect of liquid water's high SHC and bring about a reduction in temp range, which does not seem possible by any means in Aurora C# as things stand but might be worth having as you could use the hydrographic extent >20 to buffer the temperature range and reduce fluctuation due to eccentric orbit, at the cost of max population supportable. Alternatively you could invent a TN Reductium gas to do the same but since the hydrographic model already exists I just thought it would be a more interesting suggestion to make the most of it.