In a recent game, I had black, and reached the following position:

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[go]$$Bc Position at move 12
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . , X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . 2 1 . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . . . O 3 X . . . O . . . . . , . . . |
$$ | . . . . O X . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

I played D7, which Leela hated, basically telling me from after on I had no chance to win, until one point much later in the game where my opponent blundered and gave me a chance. Instead, I was supposed to play C4, leading to the following joseki:

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[go]$$Bc
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . , X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . a . . . . . . . . . . . . . . . |
$$ | . . 7 . . . . . . . . . . . . . . . . |
$$ | . 9 O X . . . . . . . . . . . . . . . |
$$ | . 8 O X . . . . . . . . . . . . . . . |
$$ | 0 3 1 O 5 X . . . O . . . . . , . . . |
$$ | . 4 2 6 O X . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

After , Leela suggests a, while Josekipedia suggests the immediate connection at D7 and says, "Black is thick. Good for Black." [Technically, Josekipedia has the colors switched.]

Why is it good for Black? It seems to me White would just play someplace like F6 or K6 and Black would get nothing out of the thickness. I tried analyzing with Leela, but it just gave a bunch of long variations that I don't understand.

compare to the game ...

big difference on the left side.

As Gomoto said, the easiest way is to compare the two results.

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[go]$$Bc White corner has an open way to the left side
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . , X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . a . . . . . . . . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . . . O X X . . . O . . . . . , . . . |
$$ | . . . O O X . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

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[go]$$Bc White corner is sealed off from the left side
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . , X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X X . . . . . . . . . . . . . . . |
$$ | . X O X . . . . . a . . . . . . . . . |
$$ | . O O X . . . . . . . . . . . . . . . |
$$ | O . . O X X . . . O . . . . . , . . . |
$$ | . O O O O X . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Your complaint in the second one was that white might be able to play a move like a and make black's influence "useless". Well, that equally well applies to the first one. The second one is still clearly better than the first one because white's corner has still been sealed off.

And if you don't add the last black move (D8) in lightvector's first diagram, there's a weakness :

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[go]$$Wc
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . , X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . 3 . . . . . . . . . . . . . . |
$$ | . . . 1 2 . . . . . . . . . . . . . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . . . O X X . . . O . . . . . , . . . |
$$ | . . . O O X . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Black has wrap-around thickness. You can't destroy the influence of wrap-around thickness with a move or two. Wrap-around thickness is huge.

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

Visualize whirled peas.

Everything with love. Stay safe.

maybe it helps you if we swich the order of moves a little(tewari):

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[go]$$Bc
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . , . . . . . , X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . b 1 a . . . . . . . . . . . . . . . |
$$ | . 9 O X . . . . . . . . . . . . . . . |
$$ | . 2 O X . . . . . . . . . . . . . . . |
$$ | 0 7 5 O 3 X . . . O . . . . . , . . . |
$$ | . 8 6 4 O X . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

narrative: black plays a hane at 1, but instead of cutting or haneing back or protecting sensibly, white plays the riddiculus move at 2. black then goes to exchange 3 for 4 in sente as you did. By means of exchanging 5-8 in sente (of course with this move order ehite would have to answer that, but he does so the exchanges are not bad) which doesnt lose points he even gets the hane at 9 in sente(white answers at 10) which prevents white from later getting b in sente, before coming back at a to protect his first hane. Do you see now, why this must be better for black than just extending?

The tewari analysis makes some sense (for once). One of the problems I had is that I saw a weakness at E4 that I felt I had to fix, and couldn't let White push through or get the cut at F5.

Imagine 2 choices:
- You get $100
- You get $500, but a criminal/the government/your opponent might steal $100 (leaving you with $400).
Which do you prefer? When stated like this, I imagine most would choose the latter. However, some people focus more on their potential loss and what the other gains ("I don't want the government to take my money", "I don't want my opponent to negate my thickness") so can end up biting their nose to spite their face.

The problem I've always had is that it's a case of a bird in the hand being worth two in the bush. Strong players claim thickness is good, but no matter what I try I've never been able to make it work for me.

So to belabour the money analogy, a thickness versus territory scenario like you describe would be:
- You get $100 cash
- You get shares which most people value at $200.
But you have a history of bad investments so plump for the cash.

But that's not really applicable here: with the sequence you chose you get thickness and white gets territory. With the suggested sequence (let's assume white doesn't resist with the good empty triangle, I doubt your kyu opponents will) you get better thickness and white gets less territory and is cut off from the left side. Any bad things that can happen to your thickness in this suggested sequence could also happen with your game line. So your choice is between 3 birds in the bush or 2 birds in the bush. If you want 1 bird in the hand then don't approach the 4-4 corner but dive into the 3-3 point like AIs do these days! (actually I don't recommend this for you).