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.

Answer 2

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

You are absolutely right that White 2 is a mistake in this diagram. After White 2 in this diagram, Black can kill White.

Instead, White should play for the seki as you've found:

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

Still, Black 1 is the correct first move for Black, since it results with Black alive in seki, whereas if Black had played any other first move, the Black stones would have died.

So, this tsumego is not about killing the White stones, it is about finding life for the Black stones.

The only defense I can give for Tsumego Pro showing you this White 2 mistake is that when solving tsumego, we're often less interested in the opponent's "best move", and more interested in the opponent's "most stubborn move" / "most complicated move" / "path of most resistance".

That is, when solving the tsumego, you have to read out every possible answer of the opponent, including this White 2 self-killing move.

But that's admittedly a poor defense in this case, and it's misleading from TsumegoPro to make White play a self-killing move like that, potentially making the player erroneously believe that they have found a sequence to kill White, whereas it should really be a seki.