So moisture decreases the temperature lapse rate (relative to the dry adiabat). My question is this: does a cloud decrease the lapse rate at the same rate as 100% humidity would decrease it? Or would it depend on cloud type (e.g. stratus vs cumulus)?
The lapse rate of temperature is ignorant of clouds, it only cares that either $e/e_s < 1$ (dry) or $e/e_s >= 1$ (moist), where $e$ is vapor pressure and $e_s$ is saturation vapor pressure. A cloud is saturated (and in places supersaturated) and so the lapse rate will be the same as saturated air outside of a cloud.
The theoretical parcel at 100% humidity when subsequently lifted (in the presence of cloud condensation nuclei (CCN)) will itself experience nucleation and condensation and become a cloud. In this respect there can be no difference between the saturated parcel outside of a cloud and a saturated parcel within a cloud, both will experience the same lapse rates.
With that said, some types of clouds will cause a difference. Consider a parcel at its level of free convection (LFC) within a deep convective cloud and a parcel far from the cloud, saturated in clear air at the same height.
- The parcel in clear air when lifted will cool at the moist lapse rate but entrain the surrounding dry air and the result will be less than moist adiabatic. The entrainment will also limit the vertical extent of the parcels buoyancy.
- The parcel in the deep cloud will be saturated in a moist environment and will rise and cool at the moist adiabatic rate. As it is inside the cloud, we'll say entrainment is not important. This parcel will also rise high enough for ice processes to become important and will experience additional warming due to the enthalpies of sublimation and fusion (latent heat released from vapor->solid and liquid->solid transitions).
In this perspective, the parcel in the cloud to have a different temperature profile than the out of cloud parcel. This isn't because of any difference in the moist lapse rate between the parcels or even how the thermodynamics work out, but rather because of the dynamic effect of entrainment.
Another factor to consider are the radiative properties of clouds vs clear air. Clouds will absorb terrestrial longwave radiation and warm, and the water vapor in humid air will do the same thing. Also on the periphery of the cloud where entertainment and evaporation is taking place, there will be some cooling from the phase change. I don't have back of the envelope estimates for either of these effects, and I feel they are probably tangential to the question asked if we limit consideration of the cloud parcel to a parcel well inside of a deep cloud.