The ERA Interim reanalysis is made by ECMWF and the other reanalysis is made by NOAA. Reanalyzes of the past are made by running global climate models and these differ because NOAA and ECMWF use different models and perhaps even slightly different data. The models on the other hand can be very different for example on implementation of data assimilation, parametrization and even grid cell size.
All of these variables are vertically integrated. The ERA variables in the order you mention them contain:
-Total column water: all water
-Total column rain water: all liquid and solid water in droplets large enough not to be considered cloud droplets (my guess)
-Total column water vapor: all gas phase water
-Total column cloud liquid water: all liquid cloud water
The NOAA variables:
-Cloud water content: all liquid and solid cloud water
-Precipitable water content corresponds to Total column water
From these the Total column water vapor corresponds the Vertically integrated specific moisture. None of these is the divergence of the total column water vapor phase that you need. Although the paper you are referring approximates precipitation from moisture divergence, you could use just the Total column rain water and no approximation is needed (I didn't read the paper so I don't know if this is makes sense).
Edit, added sources:
The ERA variable list can be found here and you can search variables there and press them to get the definitions too. For NOAA I couldn't find a source for the definitions of these variables. If someone finds please edit this answer