What exactly state is in Model Checking
My vote is on (4) from an intuition point of view.

I just started learning this recently so take this with a grain of salt but as mentioned by D.W. the set of states can be anything you want. It is an abstract concept. However, it is natural to think of states as being valuations of the variables in the system. For example, we may think of the current state of the solar system as the current values of all variables of interest to us, such as all positions and velocities of the 8 planets in it.

Seeing it this way has another advantage. If we take the atomic propositions to be formulas of the form $x = v$ where $x$ is a variable and $v$ is a value, then the labeling function is simply the identity function: $L(S) = S$, and an atomic proposition $P$ is true in state $S$ iff $P \in S$.

The fact that the state space becomes the power set of all n-tuples of valuations of variables is not really a downside. Many of these states would simply be unreachable in any practical Kripke structure. As for your concern about states being "nested", first of all note that $S_1 \subset S_2$ means that they are actually different states. Secondly, if $S_1 \subseteq S_2$, then you have the property that anything true in $S_1$ is also true in $S_2$!

In fact, as far as I have seen, this is the interpretation used in many of the practical uses of Kripke structures that I have seen, such as in model checking in the context of formal verification.