This question is related to this this older question.
Let $M$ be a smooth manifold and $G$ be a group of diffeomorphisms of $M$. In page 35 of Introduction to Lie foliations and Lie groupoids, Moerdijk and Mrčun call an $S\subset M$ a $G$-stable subset of $M$ when it is connected and given $g\in G$, either $S\cap gS=S$ or $S\cap gS=\emptyset$. Just after the definition, they point out that the $G$-stable subsets of $M$ are precisely the components of $G$-invariant subsets of $M$.
I am having a hard time to understand the implication, i.e., how a $G$-stable subset would need to be a connected component of a $G$-invariant subset of $M$. For instance, consider $M=\mathbb{C}$, $G=\mathbb{S}^1$ and $S=\{1\}$, where $G$ acts on $M$ by rotations around $0\in M$. In this case, $S$ is $G$-stable yet it is not a connected component of its group-action closure, $gS=G$.
In this context, don't we have to modify a little the definition of $G$-stable subsets only to include open subsets of $M$ or restrict it to finite subgroups of diffeomorphisms of $M$?