I like your answer the best.

I’d add that server farms sprung up to cluster network services around high speed network cables; they allow individuals to locate their services somewhere that others can access them easily.

Data centres traditionally were run by single organizations to centrally store their data — where a server farms was about providing services in the most accessible way, data centres were about efficiently storing really large data sets in a single place.

AI compute centres are really both at the same time; they store massive amounts of data AND provide an interface to that data in the same place, because the compute services need near instant access to the data sets to function properly. So you end up with heavy compute clusters and data clusters running side by side, as close to the highest speed and bandwidth network cables as possible.