Nope, it doesn't have to be open loop!
Example of such a system being used specifically for datacenters: https://blog.vantage-dc.com/2026/04/22/cooling-without-the-d...
Evaporated water is condensed, and in the process transfers its heat into another place that removes it. Another simple example is a pot of boiling water with a lid on it.
The link you cited is not evaporative cooling and a pot of boiling water with a sealed lid on it is a pressure vessel which eventually explodes.