No examples for the のs because I’m not going to attempt to sort through the vast number of instances in my translations, haha.
の1
The big surprise to me with this entry is that a bunch of things that I had been thinking of as distinct uses for の were lumped into the same grammatical item, and it made me realize that in some sense, these are all just の-adjectives? Here’s a Tofugu article that helped me understand the difference between な-adjectives and の-adjectives.
Note 1 alludes to the same sort of central point, which is that in AのB, Aの modifies B and indicates a specific member(s) of B among all the members of B.
Here’s Tofugu on how な and の are used for different kinds of noun modification:
So first, what is noun modification? Put simply, it’s using another word (often, but not always, an adjective) to give more information about that noun. For example:
きれいな人
beautiful person
In this example, the noun 人 (person) is being modified by the な-adjective きれい (beautiful), which tells us a characteristic or quality of the person (that they are beautiful). This type of noun modification is called “describing”, and is typical of both い-adjectives and な-adjectives. So, when a な-adjective is used to describe a noun, it uses な to do so.
Now let’s examine what noun modification looks like when we use particle の, rather than な:
きれいのヒント
beauty hints
In this example, the noun ヒント (hint) is being modified by きれい, just like in the first example. This time though, きれい is followed by の. If you’ve learned anything about particle の, you might know that it is used to express the relationship between two nouns. In our English translation of きれいのヒント, you can see that きれい is translated as “beauty” (a noun), rather than “beautiful” (an adjective), as it was in the previous example.
Wait — So if きれい a noun in this example, how is it able to modify another noun? As it turns out, nouns can modify other nouns, but in a different way than な-adjectives. Adjectives describe a quality or characteristic of a noun, whereas nouns label other nouns, telling us the type or category. So in our example, きれい does not describe ヒント (i.e. the hints themselves are not beautiful), it is telling us about the type of hints at hand. In other words, it helps us know that the hints are about beauty, not finance or dating. We’ll call this type of noun modification “labeling”, and の is frequently used to express this.
To quickly sum up, な is used when you want to describe a noun, and の is used when you want to label a noun.
の2
The main thing that stood out to me here was the related expressions note showing why this の was different from の1 and の3. I had totally just been assuming that this use of の came from omitting the noun, but the example given in 1c shows that it really is its own thing!
Note 2 was interesting, too. I don’t know if I’ve run into this specific ambiguity before, but it’s good to look out for.
の3
I don’t think I knew that, as explained in note 1, a nominalized sentence can occur in any position where a noun phrase can appear, except in the position of B in AはBだ. In this case, こと is used. I suppose in some way it weirdly makes intuitive sense to me, though maybe that’s just from having lots of exposure to reading monolingual definitions of things that fit that format…
I’m still trying to get a feel for times where you’d use こと and times where you’d use の. The list of examples was interesting!
の4
As usual, whenever I read about gendered usage, my impulse is to look for examples of the opposite gender using it, though sadly I don’t have a particularly extensive library of transcribed speech from male wrestlers to draw on, haha, so I couldn’t find examples of male wrestlers using の as described in note 2.
のだ
It’s fascinating to me that the previous entry said that の4 is derived from のだ/のです through deletion of だ/です, and then the related expressions note says that the sentence -final particle の4 is not the same in its meaning as の of のだ…
んです was one of those things I struggled a LOT with in the very early days, because I was seeing it all over the place but didn’t realize I should be looking under explanations of, well, のだ, if I wanted to look it up, haha. There’s info out there on the internet if you search for it directly, but I had this great grammar dictionary, you see…
Now it’s one that I don’t feel like I struggle with much in reading, but I definitely do sometimes struggle with it when outputting Japanese. I’m sure I’ve gotten it wrong at least once, either using it when I shouldn’t have or failing to use it when I should have.
I do have a fun example for this one! This is from Pro Wrestling NOAH, not Tokyo Joshi Pro Wrestling. After Katsuhiko Nakajima reunited the AXIZ tag team with Go Shiozaki, I got into a bit of an argument with one of my friends, haha, about why he did it. The company put out a trailer for their reunion match (there are Japanese subs), which basically used んだ to totally confirm my interpretation. Vindicated!! 
Here's transcription/rough translation/description of what happens in the video if you're not able to catch all of it:
After beating Go in his comeback match on 2023.05.04, Nakajima comments on his pathetic state, “何だ その無様な姿は” (which, incidentally, he also commented on back during the initial betrayal, using the same wording) and then he orders Go to stand up: “立て”. The video cuts to old footage of Nakajima breaking up their tag team on 2020.08.30 (衝撃のタッグ解散から2年と247日), where he says that Go doesn’t need him anymore, “俺のこと必要としてないでしょ?” Then it cuts back to the present, where Go takes his hand, and we hear Nakajima’s voice say that AXIZ is back, “「AXIZ」復活だ”.
Then there’s a line of text overlaid onscreen: “今はお前が必要なんだ”. So it’s saying that the reason (why Nakajima reunited them) is that Go needs him now (contrastive は! It’s contrasting 今 (2023.05.04) with 2020.08.30).
I realized that between the contrastive は and the んだ in there, it was really hard to capture the exact nuance of that sentence in a translation… And if the friend I was arguing with tried plugging that into DeepL, it gives you the basic meaning, but totally loses the explanatory aspect…
In any case, it was one of those moments where I was profoundly grateful for having learned at least some Japanese. It’s always really satisfying to be able to confirm your own interpretation like that, haha.