How to define a transfer function for a given system?
Your question is best answered by simply quoting verbatim the last section of the Bob York paper you linked to. Although here he starts talking about accuracy, it's also relevant to the whole understanding of doing Bode plots manually. It's worth noting that most engineers who use transfer functions manually rarely need to use more than the amplitude Bode plot.

In fact this is an important issue because it concerns the broader question of what we are trying to accomplish with our investigation of Bode plots. Nowadays we have the luxury of making computer-generated amplitude and phase plots in a fraction of the time it takes to draw a hand sketch. So in many respects, it simply does not make any sense to waste valuable time in trying to make a highly accurate hand sketch. If analytical accuracy is what we’re after, then the computer is a better alternative. Furthermore, it turns out that in many practical applications it is rarely important to know the phase to a tenth of a degree. Often just knowing the phase to the nearest tens place is perfectly fine!
No, the real reason to persist in learning about Bode plots is the valuable insight it gives in connecting the shape of the frequency response to the transfer function. Knowing how poles and zeroes affect the amplitude and phase ultimately allows us to approach circuit analysis from a design perspective; that is, how do we design a circuit to give a desired frequency response? In this respect, computer-generated plots are not much help. They can tell you how a circuit will perform, but they can’t tell you how to improve the circuit.
So if we keep in mind that our main goal in drawing Bode plots is usually to explore qualitative behaviour of a circuit or transfer function, then the answer to the question is yes: we can usually take shortcuts like drawing the curve through the midpoint of the phase jumps. If more accuracy is required, the simple first-order corrections that we have developed can be used to adjust the plot accordingly. If even greater accuracy is required, then a computer-generated plot is needed.

What he's pointing out is the difference between analysis, and synthesis. Analysis, figuring out what a particular circuit is going to do, is easy, which is why computers can do it, and it's why students learn to do it, again, and again, in training. Synthesis, making a circuit that does what you want, is much harder, and that's what working engineers need to do. Understanding how to draw Bode plots manually lets you understand how they are driven from pole and zero locations, which in turn allows you to drive them from simple circuit elements.
Note that there's usually more than a few ways to implement any particular circuit pole. All of them will generate the same transfer function, but only a few, or one, may be permissible for other reasons like dynamic range, impedance level, noise level, power consumption, accuracy, stability, size. You need to be able to move quickly between equivalent circuit options and their behaviour to be able to evaluate these different implementations.