How does black kill white in this Tsumego?

Question

I'm struggling with "Basic problem #46" from Tsumego Pro. It looks like this, black to play:

$$ |--------------
$$ | . O O . . . .
$$ | . . O X X . .
$$ | . O O X . X .
$$ | X X X O X . .
$$ | . . . O . . .
$$ | . O O . . . .
$$ | . . O . . . .
$$ | . . . . . . .

In the built-in "correct" solution, after black takes the vital point, white plays 2 right below, after which a, b, and c all kill white:

$$ |--------------
$$ | a O O c . . .
$$ | 1 b O X X . .
$$ | 2 O O X . X .
$$ | X X X O X . .
$$ | . . . O . . .
$$ | . O O . . . .
$$ | . . O . . . .
$$ | . . . . . . .

I don't understand why white would play 2 here, depriving themselves of a liberty. If instead it started the semeai below the three black stones, I'm not able to find a variation where black is able to avoid seki. For example:

$$ |--------------
$$ | . O O 3 . . .
$$ | 1 . O X X . .
$$ | 5 O O X . X .
$$ | X X X O X . .
$$ | 6 4 2 O . . .
$$ | . O O . . . .
$$ | . . O . . . .
$$ | . . . . . . .

(3 and 5 could also be in reverse order), or this one:

$$ |--------------
$$ | . O O 7 . . .
$$ | 1 . O X X . .
$$ | 5 O O X . X .
$$ | X X X O X . .
$$ | 3 4 2 O . . .
$$ | 6 O O . . . .
$$ | . . O . . . .
$$ | . . . . . . .

If black 3 at 4 then black is dead. If black 5 doesn't happen (either on the 3rd or the 5th move), then either:

$$ |--------------
$$ | a O O 3 . . .
$$ | 1 5 O X X . .
$$ | . O O X . X .
$$ | X X X O X . .
$$ | 6 4 2 O . . .
$$ | . O O . . . .
$$ | . . O . . . .
$$ | . . . . . . .

(or 5 at a, or 3 and 5 reversed once more) and black is dead, or:

$$ 5 anywhere but a, b, or c
$$ |--------------
$$ | a O O 3 . . .
$$ | 1 b O X X . .
$$ | c O O X . X .
$$ | X X X O X . .
$$ | . 4 2 O . . .
$$ | . O O . . . .
$$ | . . O . . . .
$$ | . . . . . . .

and white plays c, killing black. If 3 and 5 are on two of a, b, and c in the last diagram, black is dead once more.

What variation am I missing where black can kill white if white plays this way?

Answer 1

Seems Black has to take the 1-2 vital point to avoid White making 2 eyes. If White begin to deprive Black's liberty from outside, then Black get connected by taking on 1-3 and White deprives outside again. After that, both Black and White have same 3 liberties, but 2 of them are in common used. This becomes seki and it should be the best result for White.