Having started a small avalanche, what does one do with an undefended cutting stone?

Question

Having just looked at the small avalanche (nadare / konadare) in 38 Basic Joseki I tried it out as White in a 2-stone game against GNU Go and was surprised that it left out the move at a, immediately playing 9 although Ishida’s Dictionary says that after Black 6 the moves 7 – 8 – a – b are the only possible order.

$$Bc Up to the deviation from joseki
$$ ........-
$$ ....6...-
$$ ....12..-
$$ ..9.34..-
$$ ...5O7a.-
$$ ...b8...-
$$ ........-
$$ ---------

Is Black 9 indeed an error, and if so how should White punish Black? Please explain why your answer is ideal and suggest any general principles that apply. I suspect that one should capture with a, though that does make Black b sente.

How the game actually continued

In case it is relevant or of interest, though I am only asking about the joseki, here is the continuation. I was playing a little hurriedly and also was not aware that 9 was an error, and continued with:

$$Wcm10 17 connects at 14, 20… in next diagram
$$ . . . . . . . .| 
$$ . . . 7 O . . .| 
$$ . . 1 6 X O . .| 
$$ . 9 X 5 X O . .| 
$$ . . 0 X O X . .| 
$$ . . 4 3 O 2 . .| 
$$ . . . . . . . .| 
$$ ---------------+

after which both sides created moyos:

$$Wcm20 Move 29 followed by circled, then squared and finally triangled stones
$$ +-------------------------------------+
$$ |. . . . . . . . . . . . . . . . . . .| 
$$ |. . . . . . . . . . . . . . . . . . .| 
$$ |. . . Y Q . . . . 4 . . 6 . . . . . .| 
$$ |. . X , O . . O . , . . . . . X . . .| 
$$ |. . . . . . . . . . . . . . . . . . .| 
$$ |. . . X . . . 5 . . . . . . . . 0 B .| 
$$ |. . . . . . . . . . . . 9 . . 7 W W .| 
$$ |. . . . . . . . . . . . . . . . . . .| 
$$ |. . 8 . . . . . . . . . . . . . . . .| 
$$ |. . . , . . . . . , . . . . . , . . .| 
$$ |. . . . . . . . . . . . . . . . . . .| 
$$ |. . . . B . . . . . Q . . . . . . . .| 
$$ |. . . . . . . . Y . . . . . . . . . .| 
$$ |. . . . . . . . . . . . 3 . O O . . .| 
$$ |. # . . . . . . . . . . . O X X O . .| 
$$ |. @ # X . . . . . , . . O X X X O . .| 
$$ |. @ @ @ # # # . . Q . . 1 X X O X . .| 
$$ |. . . . @ @ . . . . . . 2 X O O X . .| 
$$ |. . . . . . . . . . . . . . . . . . .| 
$$ +-------------------------------------+

Answer 1

9 looks like a severe mistake to me. White may simply capture the single stone and be satisfied, like this:

$$B Simple punishment
$$ ........-
$$ ....Ob..-
$$ ....XO..-
$$ ..1.XO..-
$$ ...XOX2.-
$$ ...cOa..-
$$ ........-
$$ ---------

After this, if black c, white may answer at a, or may tenuki. Tenuki is reasonable, because the black followup does not threaten white, only captures two stones and ends in gote.

(If W2 instead at a, which is sometimes a good move, then black has a cut at b. This cut is worth more than the black descend at c.)

In other words, the cut at b works a lot worse than in the joseki for black. At the same time, black is not strong (e.g. the point you played at in the game remains a weakness). Territory is white's, too. This is not an even result.

As for why B1 was played at all: It is indeed a shape point, because it makes black a work (to capture the two white stones). In some other joseki that look (vaguely) similar, it is actually the correct move.

The underlying reason why the order was wrong becomes visible if we compare with the joseki move. If black plays at W2 instead of B1, then it is significantly harder to capture him in the corner (four liberties instead of two). That is why, in the joseki, white will need to offer a trade, losing two stones but capturing the stone at B1. (There is a ladder involved etc., but those are details.)