The S Parameters are defined in terms of a circuit within a closed box. Snm is the amplitude of the signal coming out of port n relative to the signal that went into port m. Thus, S21 is the signal that comes out of port 2 relative to what went into port 1. For a two port network that is the transmitted signal. S11 is what comes out of Port 1 relative to what goes into Port 1, so it is the reflected signal. All ports are assumed to be terminated in a matched impedance for whatever you use as the calibration impedance. This means there are no reflections off the generator or the external loads.
Some textbooks explain this more clearly than others.
One important thing to note is that for passive networks, i.e. no amplifiers or other sources, S21=S12.
I do a lot of analysis of N port networks for coupled cavity systems. These are to understand the properties of rf linear accelerators, which are nothing more than a type of bandpass filter. I find the chain matrix notation more useful than S Parameters. The chain matrix, sometimes called the ABCD matrix, relates the voltages and currents at one set of terminals, the input, to the voltages and currents at another set of terminals. For a two port network with voltage V1 and current I1 a the input and V2 and I2 at the output this gives the circuit equations V2=A*V1+B*I1 and I2=C*V1+D*I1. Note that the conventional definition of the current is that I1 is into the network and I2 is out of the network.
Given the voltage and current at a terminal the incoming wave amplitude a and outgoing wave amplitude b can be defined as a=0.5*(V/sqrt(Z0)+I*sqrt(Z0)), and b=0.5*(V/sqrt(Z0)-I*sqrt(Z0)). The scaling isn't important since the S Parameters are ratios of outgoing waves to incoming waves, but you can set it so the sum of all a^2 equals the total power input.
There is another way to look at networks called the impedance matrix which relates all terminal voltages to all terminal currents. This is what you usually get by applying Kirchoff's law to a circuit.
The point of the above is that you can use ordinary circuit analysis to a network and calculate the S Parameters to develop an understanding of how they relate to the parameters of an ordinary circuit, such as an inductor, capacitor, resistor or resonator.
This can involve a lot of messy algebra, which can easily get screwed up by mistakes. I find Mathematica to be a powerful tool, although sometimes it takes some effort to get an algebraic result in a "neat" form.
The NanoVNA is a pretty amazing instrument for its price. I think it is good not to confuse it with a laboratory instrument, but it is a great learning tool. It also has an immediate application that is useful to Radio Amateurs, understanding antennas and filters.
I have enjoyed following the discussions in this forum.
73 de K9GXC, Jim
