I think they key point here is you need to consider the pros and cons in the local continuations if black and white play first, and then look at the global position to see, given the expected flow of the game, who is likely to play first locally and who that's good for.
So my first thought is I don't want to keep on pushing from behind as that increases the strength of the black wall (which looking globally is not strong, white has a pincer stone so likely wants to attack it), but jump ahead. So there's the 3rd line move which has this obvious weakness (maybe don't need to play 3-4, but if you don't maybe white would capture on other side and let you into corner which is a big loss, but I did it to make threat on 3 stones obvious):
- Click Here To Show Diagram Code
[go]$$Bcm1
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . 4 3 . . . . . . . . . . . . . . . . |
$$ | . . O X . . O . . . . . . . X . . . . |
$$ | . . O X . . . . . , . . . . . , X . . |
$$ | . 8 O X . . . . . . . . . . . . . . . |
$$ | . 6 5 1 . . . . . . . . . . . . . . . |
$$ | . 7 2 . . . . . . . . . . . . . . . . |
$$ | . . O . . . . . . . . . . . . . . . . |
$$ | . a . . . . . . . . . . . . . . . . . |
$$ | . . X , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . X O O . . . . . . . . . . . . . . . |
$$ | . X X O . . . . . , . . . . . O . . . |
$$ | . X O . . O . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ----------------------------------------[/go]
So this means white's one point jump out to the centre (if white is next to play in the area) has a classic cutting tesuji weakness (maybe 2 at 6 directly more stylish, but W might resist). Also
a above is sente, but you don't want to play it too early as it fixes the jump weakness, but might be relevant for c8 gap:
- Click Here To Show Diagram Code
[go]$$Wcm1
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O X . . . . . . . . . . . . . . . . |
$$ | . . O X . . O . . . . . . . X . . . . |
$$ | . . O X . . . . . , . . . . . , X . . |
$$ | . O O X . . . . . . . . . . . . . . . |
$$ | . O X X . . . . . . . . . . . . . . . |
$$ | 7 X O 8 . . . . . . . . . . . . . . . |
$$ | 5 2 O 6 1 . . . . . . . . . . . . . . |
$$ | . 3 4 . . . . . . . . . . . . . . . . |
$$ | . . X , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . X O O . . . . . . . . . . . . . . . |
$$ | . X X O . . . . . , . . . . . O . . . |
$$ | . X O . . O . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ----------------------------------------[/go]
Compare this to jumping following the kosumi, black's typical attempt to cut the shape is peep and cut at waist, but it doesn't work as white's corner stick is now 4 not 3 stones long (and white is safe even if black 2 is connected):
- Click Here To Show Diagram Code
[go]$$Wcm1
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . 3 2 . . . . . . . . . . . . . . . . |
$$ | . . O X . . O . . . . . . . X . . . . |
$$ | . . O X . . . . . , . . . . . , X . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . 6 7 X . . . . . . . . . . . . . . . |
$$ | . O 8 4 . . . . . . . . . . . . . . . |
$$ | . 9 O 5 1 . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . X O O . . . . . . . . . . . . . . . |
$$ | . X X O . . . . . , . . . . . O . . . |
$$ | . X O . . O . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ----------------------------------------[/go]
So if white wants to play next in the area to jump out, the kosumi is better (though there's also e13 knight jump to consider which stops the tesuji of diagram 2, but leaves another cut).
How about if black first? If black tries to cover the 3rd line move on 4th line (5th at
a is flimsy shape with an obvious gap) white can maybe push out though black has that cutting tesuji on 2nd line to make liberty troubles (maybe corner hane first) but white might allow the cut and develop the outside group:
- Click Here To Show Diagram Code
[go]$$Bcm1
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O X . . O . . . . . . . X . . . . |
$$ | . . O X . . . . . , . . . . . , X . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . 4 3 X . . . . . . . . . . . . . . . |
$$ | . 5 O 2 . . . . . . . . . . . . . . . |
$$ | . . O 1 a . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . X O O . . . . . . . . . . . . . . . |
$$ | . X X O . . . . . , . . . . . O . . . |
$$ | . X O . . O . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ----------------------------------------[/go]
Or maybe white would look at the whole board and tenuki, extending that lone stone on the top is clearly an important place. Maybe black spends another move to connect, and white extends again? Or 4 somewhere else like lower left or right side?
- Click Here To Show Diagram Code
[go]$$Bcm1
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O X . . O . . 2 . . 4 . X . . . . |
$$ | . . O X . . . . . , . . . . . , X . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . . O 3 . . . . . . . . . . . . . . . |
$$ | . . O 1 . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . X O O . . . . . . . . . . . . . . . |
$$ | . X X O . . . . . , . . . . . O . . . |
$$ | . X O . . O . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ----------------------------------------[/go]
Compare if the same idea happens with kosumi (black's reinforcement with 3 is a bit different):
- Click Here To Show Diagram Code
[go]$$Bcm1
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O X . . O . . 2 . . 4 . X . . . . |
$$ | . . O X . . . . . , . . . . . , X . . |
$$ | . . O X . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . . . |
$$ | . . O 1 . . . . . . . . . . . . . . . |
$$ | . . 3 . . . . . . . . . . . . . . . . |
$$ | . . X , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . X O O . . . . . . . . . . . . . . . |
$$ | . X X O . . . . . , . . . . . O . . . |
$$ | . X O . . O . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ----------------------------------------[/go]
How to judge these 2 results? I think they are both globally good for white who got to settle nicely on the top side. How about the left side shape? c10 got an extra stone in the 2nd, so maybe c8 area stronger, but also the wall is not so thick and has a cut/peep. Top one the black wall is thicker and has double hane if white tries to crawl along left side, so is probably a bit worse for territory but better thickness to top side but not left side.
So I think the key point is white is happy with either of these results (though perhaps one subtly better), so black won't allow that and will pincer on the top side, which means the fact kosumi has better shape for white to come out later is most important. Also from an efficiency standpoint c13 makes c12 look a bit odd.