Naming is tricky. There are 2 sorts of count which we would need to qualify: docs-in-set and docs-with-term and we provide these stats for 2 different types of set: _subset_ (agg bucket or result set) and _superset_ (index).
The "superset-docs-with-term-count" is Lucene's "docFreq" which may be a convention used elsewhere in es APIs? 
I had tried to hold back from referring to "doc" in these labels as I thought in some cases we might want to count frequencies of something else other than docs (e.g. use case 2 of the credit card fraud scenario where we want to count-distinct related entities not volumes of transactions).  That may be a demanding scenario so counting docs may be what we stick with.