In Fundamental Principles of Go, Yilun Yang puts big opening moves in four classes. Now, each class contains a few possible move types. For example:
So, the possible moves are numbered: are they intended to be usually applied in this order?

No.

And don't forget these are not hard and fast rules but general guides: for example in the very popular mini Chinese a 4-4 is approached (3rd class) and answered and then a side move played before a 3-4 is enclosed (2nd class).

Yeah I think his only "hard" point was that approaching a 4-4 (third class) instead of taking an empty corner (first class) is likely to be an error. Or making a gote 2-space or 3-space extension not necessary for eyespace (fourth class) versus approaching a 3-4 stone (second class). It's probably possible to construct board positions where these aren't errors, but in general they seem like good general advice to me so long as you remember he's talking a 3-4 stone alone in a corner, not a 3-4 stone as part of a Chinese or Mini-Chinese formation or whatever.

Played professionally >=32 times, by players such as Lee Changho, Cho Chikun, Nie Weiping and O Meien (ok he's a bit wacky). White normally (50%) answers with the knight's move. It's a lot rarer for white to approach a 4-4 on move 4 after black takes a second corner, presumably because then black can get 3 corners, whereas when black approaches for move 3 if white takes an empty corner black can take the 4th if he wants, or double approach. Compare that with white's approach for move 3 against a 3-4 in the Shusaku opening (particularly popular in no-komi Go).

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I think he was primarily aiming at a kyu audience where the advice isn't bad. My wording above is terrible, I meant "likely an error" in the sense if you're a 5k or a 10k doing it rather than a pro as DDKs and weaker SDKs are who this is aimed at. You can tell kyu players to generally play in one of the open corners there for Black 3 and they'll be fine or you can start giving exceptions and risk confusion. You are much more likely to go wrong approaching the 4,4 than you are taking the corner if you need lists to help your move choice in the opening no?