From 4-4, AI like to follow up with a knight move enclosure. It inhibits a 3-3 invasion and it also inhibits an approach move.
But very early in the opening it can be a bit slow and it can be somewhat hard to choose the direction of the knight move.

To prevent a knight move enclosure, there are 3 options: approach or 3-3 invasion. With few stones in the general area, LZ adheres very similar winrates to those options.

Very early in the opening, it can be a bit difficult to choose which approach is best (and the opponent can even ignore the approach, LZ does that a lot and pros seem to follow that trend).

A 3-3 invasion kind of puts he onus on the opponent to commit to a direction for breaking the local symmetry, committing to making good use to the outside wall in the direction that he chooses.

Why do you expect balance / consistency from a simple proverb? Although this proverb has plenty of times it does apply and is a useful guide to thinking, particularly where the opponent playing at his key point is high up your policy network but you playing there is not (e.g. https://senseis.xmp.net/?SlappingTesuji), there's also plenty of times it's not[*] the exact move and thus would be more accurately but less pithily phrased as "The opponent's good local area is your good local area". It's pretty obvious that a 4-4 and then 3-3 enclosure is inefficient.

* for example:
Black's tesuji to capture

Click Here To Show Diagram Code

[go]$$B
$$ . . . , . . . . . ,
$$ . . . X X X . . . .
$$ . . X O O O X . . .
$$ . . X . 1 . X . . .
$$ . . . . O . . . . .
$$ . . . . . . . . . .[/go]

A bamboo joint is usually more efficient as better for outside, though if liberties are important pole connection can be better

Click Here To Show Diagram Code

[go]$$W
$$ . . . , . . . . . ,
$$ . . . X X X . . . .
$$ . . X O O O X . . .
$$ . . X . . . X . . .
$$ . . . 1 O . . . . .
$$ . . . . . . . . . .[/go]

Or whilst there are many instances of the proverb in action when killing / living with a group (e.g. nakades, vital eye points), there are also many instances where the killing move is not the same as the living move (e.g. L+1 groups).

I have the feeling that approaching a corner is not necessarily identical value as preventing the same approach: their followups (or the followups to followups if ignored) differ. An approach has a higher chance of being sente than an enclosure (even if not immediately), so it may be a bit behind in dropping temperature. (Edit: This difference may be even larger for invasions.)

After the 4-4 player has made the knight enclosure, bots and post-AI humans have a noticeable difference from pre-AI huamns in the likely follow ups for the opponent (at least this is my impression from watching pro/AI games, I've not done a historical pattern search). The knight's enclosure has been pretty common in top pro play since at least the 90s with Korean style. If white played next in the area it was common to play one of the 2nd line approaches at a/b, the thinking being (afaik) if you played the regular knight approach when black had already backed off then they could kick and pincer and that was generally bad for you. However, bots will often approach at 2 below anyway, apparently not fearing the kick and pincer. Black will quite often tenuki anyway, and it reverts to white approaching first, but black got to choose which direction white approach from so could presumably make it the bad one. So why doesn't white fear the kick? From playing around with Lizzie I think the reasons can be
1) white won't extend but tenuki or maybe play the (formerly) "crude" 2nd line hane and atari on top then tenuki
2) white will extend because that's locally a bad shape for black (bad aji at 3-3 and the 2 stones aren't under as much pressure as if black had kicked first and then played the one point jump on 4th line instead of knight to 3rd line -- all stuff we already knew before AI, but I think bots have changed the balance of judgement of these factors) and then if black pincers could
a) tenuki and treat 2 stones lightly for now, not easy to kill aji-free in one move
b) or answer and keep fighting, 2 space jump is a nice haengma for example aiming at corner shenanigans.

Click Here To Show Diagram Code

[go]$$B
$$ +----------------
$$ | . . . . . . . .
$$ | . . . . . . . .
$$ | . . . . . 1 . .
$$ | . b . X . . . .
$$ | . a . . . . . .
$$ | . . 2 . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .
$$ | . . . . . . . .[/go]

I accept that a mutual good point needn't be on the exact same spot. Your example fits very well in that paradigm. The function is the same (cut/connect) the vital point is different.

With the situation in the corner, there is a difference in function

Click Here To Show Diagram Code

[go]$$B
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . X . . . |
$$ . . . . 1 . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ --------------------[/go]

Develop and stabilize

Click Here To Show Diagram Code

[go]$$W
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . X . . . |
$$ . . . . 1 . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ --------------------[/go]

Block development, threaten stability

Click Here To Show Diagram Code

[go]$$W
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . X . . . |
$$ . . . . . . . 1 . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ --------------------[/go]

Undermine stability

Click Here To Show Diagram Code

[go]$$B
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . X . . . |
$$ . . . . . . 1 . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ --------------------[/go]

Confirm stability.

This move is not exactly where White would invade but it has the same function. The corner enclosure has a function of development.
Yet this is not a popular bot opening move either.

The approach and the invasion have a different function. Yet their counterpart would be exactly the same. I have a philosophical/logical problem with that.

My favorite example is this one.

Click Here To Show Diagram Code

[go]$$B The opponent's good move is my good move!
$$ . . . . . . O X . . .
$$ . . . . . . O X . . .
$$ . . O . . . O X . . .
$$ . . . . O O X . . . .
$$ . . O . . X X . X . X
$$ . . O X X . . . . . .
$$ . . O . . . . . . . .
$$ . . . . . 1 . . . . .
$$ ----------------------[/go]

More seriously, Knotwilg - the differences in "function" you bring up seem like a weird way to reason about things. If making the 3-6 enclosure from a 4-4 point is sufficient to make white not want to invade 3-3 any more for at least quite a while longer into the future, and is more efficient overall (even after taking into account that 3-3 is still possible and another move would be needed in the future to entirely remove the possibility) , it seems like that move qualifies as opposing the function of the 3-3.

Why would I singlemindedly play a move to prevent solely what one might subjectively perceive as the "function" of move the opponent might play when I can prevent it (at least temporarily) *and* do something else simultaneously?

Moreover, as I mentioned earlier, any time moves have large probing value, that should also break the need even for the "overall area's" urgency to match up for the two players playing in that area.

Click Here To Show Diagram Code

[go]$$B
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . X . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ --------------------[/go]

Confirm stability plus sleeve tesuji to maintain efficiency

Click Here To Show Diagram Code

[go]$$B
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . X . . |
$$ . . . . . . , . . . |
$$ . . . . . . 1 . . . |
$$ . . . . . . . . . . |
$$ . . . . . . . . . . |
$$ --------------------
$$ {AR P4 Q5}[/go]

I think that the bots are showing us that The opponent's move is my move is even less the case than we thought. OC, it is true in the weak sense that I can prevent the opponent from playing on a point by playing there myself. And it is true in the strong sense that there are often points that are important for each player.

Bots often prefer to approach an opponent's stone closely, even with an attachment or diagonally adjacent play, but the same play by the opponent would be over concentrated and inefficient. That's often the case with the 4-4 stone. Also, I have noticed, looking at the Elf review sgf files, that Elf likes the 3-3 invasion against not only the 4-4, but also the 5-4 and 5-3. All of these 3-3 invasions were not commonly played by humans before the bot era. Also, bots pincer the 5-3 approach to the 3-4 stone less than humans did, but also very seldom extend from the approach stone as the third play in the corner, but instead play the press very, very frequently.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

As Bill said, moves played in close proximity (attachments, shoulder/armpit hits) to the opponent's stones (which bots tend to like) are unlikely to be your move too due to inefficiency:

Click Here To Show Diagram Code

[go]$$W Reduce moyo
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . . O . . X . . X . . . . . . X O . . |
$$ | . . . O . . . . . 1 . . . . . X O . . |
$$ | . . . . . . . . . . . . . . . X . . . |
$$ | . . O . . . . . . . . . . . . . O O . |
$$ | . . . . . . . . . . . . . . X . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |[/go]

Click Here To Show Diagram Code

[go]$$B Grow moyo
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . . O . . X . . X . . . . . . X O . . |
$$ | . . . O . . . . . , . . . . . X O . . |
$$ | . . . . . . . . . . . . . . . X . . . |
$$ | . . O . . . . . . 1 . . . . . . O O . |
$$ | . . . . . . . . . . . . . . X . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |[/go]

Click Here To Show Diagram Code

[go]$$B Inefficient grow moyo
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . . O . . X . . X . . . . . . X O . . |
$$ | . . . O . . . . . 1 . . . . . X O . . |
$$ | . . . . . . . . . . . . . . . X . . . |
$$ | . . O . . . . . . . . . . . . . O O . |
$$ | . . . . . . . . . . . . . . X . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |[/go]