Understanding the construction of stochastic processes
This construction you describe is not fully general. In fact it characterizes strictly stationary time series. You see that it's shift-invariant. This operator $S$ is essentially a shift operator.

For comparison, here's the usual definition of, let's say discrete-time, processes:

Definition A stochastic process is a sequence $\{ X_t \}$ of Borel measurable maps on a probability space $( \Omega, \mathcal{F}, \mu )$.  

Now for what you're describing, you have a fixed Borel measurable map $X: \Omega \rightarrow \mathbb{R}^n$. It's the underlying measure that is evolving according to $S$. The map $S$ induces a new "push-forward measure" (in measure-theoretic parlance) on $\Omega$ by just taking preimages: define a measure $\mu_S$ by

$$
A \in \mathcal{F} \stackrel{\mu_S}{\mapsto} Pr(S^{-1}(A)).
$$

So the random vector $X: ( \Omega, \mathcal{F}, \mu_S) \rightarrow \mathbb{R}^n$ is $X \circ S$ by construction. They induce the same push-forward measure on $\mathbb{R}^n$. Do this with $S^t$ for each $t$ and you have your time series.

As for your question about $\omega$, inspecting the proof for the other direction should clarify this---i.e. any strictly stationary time series must necessarily take this form for some $( \Omega, \mathcal{F}, Pr)$, $X$, and $S$. 

The basic point is that, from a general point of view, a stochastic process is a probability measure on the set of its possible realizations. This is seen in, for example, Wiener's construction of Brownian motion; he constructed a probability measure on $C[0, \infty)$. So in general, an $\omega$ is a sample path and $\Omega$ consists of all possible sample paths.  

For example, take the two processes you named above. They are strictly stationary, if let's say the innovations are Gaussian. (Any covariance-stationary time series driven by Gaussian innovations is strictly stationary.) The construction would then start by taking $\Omega$ to be the set of all sequences, $\mathcal{F}$ the $\sigma$-algebra generated by coordinate maps, and $Pr$ the appropriate measure. For the white noise process (2), $Pr$ is just a product measure on an infinite product. 

Reference This characterization/construction by shift of strictly stationary time series is mentioned in White's Asymptotic Theory for Econometricians.