Came across this notation today which I've never seen before. It was used with no explanation. For the equation to make sense, I think $\textbf{1}_{\{x \in S\}}$ means, "It takes the value 1 for $x \in S$ and $0$ otherwise." I.e., it is like an indicator function. Is this correct/common notation?
You got it. Personally I know of $\mathbb 1_X$ and $\chi_X$ as notation for the indicating function of a set.