In his book on Proper and Improper forcing Shelah writes on page 89:
In Sect. 1 we introduce the property "proper" of forcing notions: preserving stationarity not only of subsets of $\omega_1$ but even of any $S \subseteq \cal S_{\leq\aleph_0}(\lambda)$. We then prove its equivalence to another formulation.
My question is how $$S\subseteq \cal S_{\leq\aleph_0}(\lambda)$$ is a generalization of $\omega_1$ since $\omega_1$ is uncountable but $$\cal S_{\leq\aleph_0}(\lambda)$$ are at most countable subsets of $\lambda$.
Also I would like to understand what is $\text{Lim}\bar{Q }$ on -5th line on that page 89.
Finally, I would like to know what is that another formulation in the snippet above.
For the first question, $\omega_1$ is a subset of $[\omega_1]^{<\aleph_1}$ (the set Shelah denotes $\mathcal S_{\le\aleph_0}(\omega_1)$), namely, any countable ordinal is a countable subset of $\omega_1$. You can check that $\omega_1$ is club in $[\omega_1]^{<\aleph_1}$, and that being a stationary subset of $\omega_1$ in the sense of ordinals and in the sense of collections of countable subsets coincide.
For the second question, $\mathrm{Lim}\bar Q$ is the countable support limit of the iteration sequence $\bar Q$. Shelah uses $\mathrm{Lim}_{<\aleph_0}\bar Q$ for the finite support limit. What $\mathrm{Lim}\bar Q$ means (i.e., what kind of supports are being considered) usually depends on context; in the page you quote, it is clear that he means countable supports.