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.