S-matrix for 2-port network

Consider an impedance $Z$ on a waveguide of impedance $Z_0$. I add a load $Z_L$ at the end of the line but I don’t think it should enter in the calculation ?

I call $1$ the left port, $2$ the right one. I can define four voltages:

$V_1^{leftarrow}$, $V_1^{rightarrow}$ are the reflected and incoming voltages on the left side of the impedance. $V_2^{rightarrow}$, $V_2^{leftarrow}$ are the transmitted and "incoming from the right" voltage. The total voltage on the left is then $V_1=V_1^{leftarrow}+V_1^{rightarrow}$, and on the right: $V_2=V_2^{leftarrow}+V_2^{rightarrow}$.

We know that current and voltages are related through (where the $-$ is is actually a matter of conventions)

$$V^{leftarrow}=-Z_0 I^{leftarrow} $$
$$V^{rightarrow}=+Z_0 I^{rightarrow} $$

My questions:

How can I compute the S matrix of this 2-port ?

What I would like to do is to naïvely say $(V_1-V_2)=Z I$. But what is $I$ here ? Do we have $I=I_1=I_2$ ? I am not so sure because we have propagating phenomenon. Is it true because we do the lumped approximation, so the impedances being "small" with respect to $lambda$ we indeed have $I_1=I_2$ ?

How to cleanly understand the derivation of $S$ matrix in this simple example ? My point is really here. I would like a super clean derivation of stuff. I am very confused when we deal with such propagation phenomenon: it is very different from non propagating electricity.