I have sent this question to someone at WRI.
Now I also want to know how/if access to various datasets (e.g. WordData[], GraphData[], etc.) affects the counts as well.
When I make the following (throwaway) function, I can see the desktop client saying it is downloading stuff from a Wolfram server.
genRanGraph[] := Module[{g0},
g0 = RandomChoice[GraphData[]];
{g0, GraphData[g0, "AllImages"]}
]
Since I have a finite set of API calls, I want to know if I can see my allotment.
Thanks in advance.