Didn’t read the links, so I don’t know what I’m repeating or…
There are many ways to make a digital converter, just like there are many ways to build a house (wood framed with stucco and drywall, bricks and mortar…).
Probably the most straight forward way is a flash converter: You have a series resistor string to ground with the input voltage at top, and at each junction a comparator—you see which ones light up, and encode that into a digital word. 256 compactors for 8-bit, 65526 for 16…no you won’t be using these for audio, gets impractical real fast. (But I used to work for a company that sold them for $500 each for an 8-bit converter for video, decades ago—real fast).
The most basic realistic converter for audio is a successive-approximation converter (based on a SAR—successive approximation register). A basic and accurate DAC is pretty simple to make, so this method leverages a DAC. A SAR ADC basically takes a digital guess, then compares the voltage output of the DAC to the input voltage. Too low, it guesses higher, and too high it guesses lower. Using a binary search, it doesn’t take that many guesses. And there’s plenty of time between samples. To get the input voltage to hold still over that time, there is a “sample and hold” step—basically charging a small capacitor quickly then disconnecting it to hold that value.
There are other variations of that, running at a higher interval rate and using a smaller DAC—all the way down to one-bit. But that’s it in a nutshell.