> On that note, and related to your clarifying remark about corelations, do you happen to know of any terms that I should take note of when considering two partitions on the same set (rather than what I'm understanding as one partition on two sets)?

In game theory, overlapping partitions are sometimes referred to as *information partitions*. Each partition corresponds to a particular player. This leads to a notion of common knowledge Robert Aumann introduces in his 1976 *Agreeing to Disagree*. The [Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/common-knowledge/#2.3) has a nice article about this.