How does multiple layer seismic reflection work? How do we distinguish arrivals from different layers and their velocities?
It is impossible to to convert one recorded time-series into structural information. As you correctly note, the equation
$$ v_i = \frac{z_i}{t_i}, \tag{1a}$$
or
$$ t_i(z,v) = \frac{z_i}{v_i}, \tag{1b}$$
is difficult to solve when you only a know one time $t_i$. That is, choosing $z_i$ to twice its value would provide twice $v_i$! If you only record time $t$, but try to turn it into two values $(z_i,v_i)$, you have an underdetermined problem.

However, if you have multiple traces, you know from the theory that a flat reflector produces a hyperbolic move-out (I believe it's the following equation, but I'm not 100% certain. Don't quote me on it!):
$$ t_i(x,z,v) = t_0 + \frac{x^2}{v_\text{stacking}}. \tag{3}$$
As you know the location $x$, knowledge of two (rather than one) traces would perfectly constrain $t_0$ and $v_\text{stacking}$! Just like the slides you provide in the link, the velocity $v_\text{stacking}$ is usually simply found by trying many values for the move-out correction velocity, and seeing which velocity value flattens the reflector best! From that knowledge, suddenly you can constrain also the depth, etc.

Reflection seismics deals with the situation that is described in the 5 slides preceding the slide you post. We have a lot of traces, and based on the assumption of hyperbolic moveouts we can perfectly obtain the stacking velocities. The slide you post approaches the problem from the other way around: what are the stacking velocities $v_\text{rms}$ you would find if you know the depth and velocity structure already? This is not necessarily the method used to get structural information from your data! I think that is the cause of the confusion..!