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The author is a muppet babbling about what they don't know.

TL;DR: here's a better guide. 90s video included if you don't want to read shit.

An extra tip is to add some baking soda (sodium bicarbonate) to the boiling water. This will increase the boiling temperature of water

For kitchen purposes, the effect of dissolved salts in the boiling point of water is negligible. For baking soda you'd need to add more than 1tbsp for each cup of water, to raise it by a whole bloody degree. And even if the effect was relevant you'd achieve the same with any other solute, even table salt.

The actual reason you might want to add some baking soda (or vinegar, or salt etc.) to the boiling water is because it denatures proteins, so if one of your eggs cracks while boiling, the white solidifies faster; hopefully sealing the crack off. This means cleaning less egg white gunk from your pot. (The shape of the boiled egg is probably ruined anyway. I generally throw those in - eat your failures so they won't haunt you.)

Once you boil the eggs for a particular amount of time (which we will discuss later), you have to rapidly cool them. This ensures that we utilize the difference in thermal conductivity between the different layers of the egg. The results is a shell that is easier to remove.

Thermal conductivity is irrelevant here. What could matter is thermal expansion: things get smaller when cold, but at different ratios, but even then I don't think thermal expansion plays a big role here.

The actual reason why "throw them into cold water" is good advice threefold:

1. because it stops the cooking process immediately, so you get a more consistent result. Carryover cooking is a thing.
2. because the egg white gets firmer, so you're less likely to rip it apart.
3. because peeling hot things is a literal pain.

For #2 and #3 you could also fridge them overnight. I don't bother, I'm usually too eager to eat them.

There is some amount of moisture trapped between the shell and the albumin. It prevents the sulfur and iron inside the yolk from forming iron sulfide, which is the grayish color we often see on over-boiled yolks.

The moisture has jack shit to do with this, but point #1: you're preventing the egg from getting overcooked. And guess what, it's cooking that breaks sulphur off the proteins! That's why, for example, you'll get green/grey yolks if you cook the eggs for too long, even if you dump them into cold water afterwards.

Time is another important factor. If you want to achieve a particular style of boiled egg you need to boil it for a certain temperature. I usually eat hard-boiled eggs and 11-13 minutes after I have added the eggs to the boiling pot works well for me. If you want to do 9, you can experiment with 9 as well. That could lead to a jammy yolk and the white being solid. It results in eggs which have good visual, Instagrammable looks.

A picture is worth a thousand words:

Taken from J. Kenji López-Alt's guide on perfect boiled eggs. Way better guide than the one in the OP, showing another way to do this: start with already boiling water, gently land the eggs into the water with the help of a spoon, wait, remove them from the water, profit.