From the Merriam-Webster Dictionary:
a number of similar things that occur together
The two uses of the term that you describe have to do whether you are trying to discover a cluster in a data set or whether you are trying to account for known clusters in a data set. The first use is what you are familiar with already, so here's a brief explanation of the second.
Many statistical tests are based on an assumption that the observations are "independently and identically distributed" (iid). That assumption, however, is often not tenable. For example you might be evaluating results for individuals who are inherently grouped in ways that might lead to outcomes being correlated within each group: students grouped within schools, patients within hospitals.
There are several ways to account for such multi-level structuring of data, discussed for example on this page. The "cluster" term that you see as an option in many regression models is one way to do that. It takes the associations of outcomes within each "cluster" into account when estimating the standard errors of coefficient estimates.
The dictionary entry cited above has two additional, self-explanatory uses of the term that are important in statistics:
a larger than expected number of cases of disease (such as leukemia) occurring in a particular locality, group of people, or period of time
a number of computers networked together in order to function as a single computing system.
