That is like someone asking how a mobile phone works and you answer by explaining how an analog wall phone works!
What you quoted from Oratory, creating an anti-phase (out of phase) signal is NOT at all the algorithm used in these advance ANC headphones. None work in analog domain to simply invert a phase of a signal and mix it. Indeed Oratory goes on to bullet list of reasons why such an idealistic, paper solution fails to produce state-of-the-art noise cancellation.
The solution used by Sony, Beats, Bose, etc. is all in digital domain using an adaptive filter. Here is the general block diagram:
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A multi-tap adaptive FIR filter is used as the heart of the system. The filter starts with unity coefficients meaning it does nothing to the signal. The filter coefficients are filled in over time using that "prediction unit." A feedback signal, e(t), instructs the system as to how far off it is from filtering the input noise component, u(t). This is a non-trivial, computationally intensive problem to solve. Various schemes are used for arriving at an answer with the most common being Filtered LMS (FxLMS). LMS stands for least mean square and is a mathematical metric of error. So by minimizing that, it is assumed that the audible perception of noise is reduced proportionally (this is not perceptually accurate but is good enough here).
Now back to the argument, the filter is programmed in real-time for the noise source in question. For the duration of adaptation (which could be a few seconds), the noise needs to be what we call in signal process, "stationary" meaning its make up does not change. If it does, then all the previous training of the filter goes out the window and the system has to start over to characterize the new source.
A stationary noise is one that doesn't change over time. A computer fan noise is such a noise. As are other mechanical sources of noise (airplane for example). Any kind of cyclical noise is a friend of ANC and its adaptation since its spectrum remains the same.
What isn't stationary is impulsive noises like a shot-gun, or someone banging something. Since the initial strike is different than the spectrum after that, the adaptation by definition can't work on the initial impulsive sound. This is why as I was testing the Sony, as my wife worked in the kitchen, on every bang as she was handling the dishes and such, noise cancellation would go away completely and then gradually come back (the tail end of underdamped system is filterable).
Anyway, the math and science here is quite complicated and I don't know that I can simplify it any more than this. If you want to learn more, search for "adaptive digital filters" and or "LMS" and you should find more information.
BTW, this area is a patent minefield. Given how popular this technology is, every random patent troll will go after you on this. This is why you don't see any detailed write up from headphone companies how their systems work. You can tell some though from their marketing material. Here is Sony:
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Dual microphone (sampling input noise and output error) is a give away as is digital processing in their VLSI.
Apple says a bit more about their Beats headphones but basically the same message:
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You can clearly read the message of adaptation as opposed to simple analog phase inversion you talked about.
Note as headphone designers, you usually just buy the DSP and software to go with it that performs this functionality. It is only companies like Sony, Apple, etc. that are big enough and have their own lawyers and patents to defend their algorithm, that you will see proprietary implementations. So someone being in headphone business does not at all arm them with knowledge of how these systems work. To them, it is just a black box and hence the ancient/lay scheme Oratory explained.