You're making a mistake, at least for the second case:

In the second case, the water ends up as rain, presumably within a few hundred kilometers of the evaporation point.

You cannot model a dry region as a closed system for the purposes of water contained. When water evaporates in a dry climate, it transports much farther than a few hundred kilometres. General circulation can transport airmasses for thousands of kilometres. In all likelihood, when the water finally precipitates it will do so in a different catchment area and/or far upstream, in an area that already has plenty of precipitation. This is why in particular hydro lakes in hot climates have such a large impact on ecology: a hydro lake is far larger evaporation than a river, due to its much larger surface area and other factors.

You're making a mistake, at least for the second case:

In the second case, the water ends up as rain, presumably within a few hundred kilometers of the evaporation point.

You cannot model a dry region (or indeed any region on earth) as a closed system for hydrological purposes. When water evaporates in a dry climate, it transports much farther than a few hundred kilometres. General circulation can transport airmasses for thousands of kilometres. In all likelihood, when the water finally precipitates it will do so in a different catchment area and/or far upstream, often in an area that already has plenty of precipitation. From there it may flow thousands of kms to different climate areas yet. This is why in particular hydro lakes in hot climates have such a large impact on ecology: a hydro lake is far larger evaporation than a river, due to its much larger surface area and other factors. When water from a dry region is gone, it can, for all practical purposes, be counted as a loss.