I think extending into the corner is better in this case, because I think white ends up too weak if he cuts. You seem to notice that yourself, which is good.
Here's a few tips on the plays in the diagrams you came up with.
$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . O X . . . . . .
$$ | . . O X X . . . . . .
$$ | . 3 O O X . . . . . .
$$ | . 1 X X O O . . . . .
$$ | . 2 . X X O . 7 . . .
$$ | . . . . 6 4 5 . . . .
$$ | . . . . . . . . . . .
$$ ----------------------
- Click Here To Show Diagram Code
[go]$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . O X . . . . . .
$$ | . . O X X . . . . . .
$$ | . 3 O O X . . . . . .
$$ | . 1 X X O O . . . . .
$$ | . 2 . X X O . 7 . . .
$$ | . . . . 6 4 5 . . . .
$$ | . . . . . . . . . . .
$$ ----------------------[/go]
This

gives white better eyeshape while protecting the cut, and also avoids forming an empty triangle.
$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . O X . . . . . .
$$ | . . O X X . . . . . .
$$ | . 3 O O X . . . . . .
$$ | 8 1 X X O O . . . . .
$$ | . 2 . X X O . 7 . . .
$$ | . . . . 6 4 5 . . . .
$$ | . . . . . . . . . . .
$$ ----------------------
- Click Here To Show Diagram Code
[go]$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . O X . . . . . .
$$ | . . O X X . . . . . .
$$ | . 3 O O X . . . . . .
$$ | 8 1 X X O O . . . . .
$$ | . 2 . X X O . 7 . . .
$$ | . . . . 6 4 5 . . . .
$$ | . . . . . . . . . . .
$$ ----------------------[/go]

is a small endgame move. It is worth only a couple of points. These moves shouldn't be played until the end of the game.
$$Bm8
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . 2 . . . . . . .
$$ | . 3 1 O X . . . . . .
$$ | . . O X X . . . . . .
$$ | . O O O X . . . . . .
$$ | . O X X O O . . . . .
$$ | . X . X X O . O . . .
$$ | . . . . X X O . . . .
$$ | . . . . . . . . . . .
$$ ----------------------
- Click Here To Show Diagram Code
[go]$$Bm8
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . 2 . . . . . . .
$$ | . 3 1 O X . . . . . .
$$ | . . O X X . . . . . .
$$ | . O O O X . . . . . .
$$ | . O X X O O . . . . .
$$ | . X . X X O . O . . .
$$ | . . . . X X O . . . .
$$ | . . . . . . . . . . .
$$ ----------------------[/go]
Instead, cutting white seems like a good move. If

, take your time to confirm that

captures the five white stones.
If white plays as below:
$$Bm8
$$ | . . . . . . . . . . .
$$ | . . 5 . . . . . . . .
$$ | . 4 3 . . . . . . . .
$$ | . 2 1 O X . . . . . .
$$ | 6 . O X X . . . . . .
$$ | . O O O X . . . . . .
$$ | b O X X O O . . . . .
$$ | 7 X . X X O . O . . .
$$ | . a c . X X O . . . .
$$ | . . . . . . . . . . .
$$ ----------------------
- Click Here To Show Diagram Code
[go]$$Bm8
$$ | . . . . . . . . . . .
$$ | . . 5 . . . . . . . .
$$ | . 4 3 . . . . . . . .
$$ | . 2 1 O X . . . . . .
$$ | 6 . O X X . . . . . .
$$ | . O O O X . . . . . .
$$ | b O X X O O . . . . .
$$ | 7 X . X X O . O . . .
$$ | . a c . X X O . . . .
$$ | . . . . . . . . . . .
$$ ----------------------[/go]
White has to [sl=TheSecondLineIsTheRouteToDefeat]push along the second line[/sl] twice before coming back with

to make two eyes. Black might now have to play

or 'b' (or the more simple but slightly worse for points 'c') to avoid a white play at 'a', which turns the situation into a rather complicated life and death situation which I don't think black can survive from.
$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . O X . . . . . .
$$ | . . O X X . . . . . .
$$ | 9 3 O O X . . . . . .
$$ | 8 1 X X O O . . . . .
$$ | . 2 . X X O . 7 . . .
$$ | . . . . 6 4 5 . . . .
$$ | . . . . . . . . . . .
$$ ----------------------
- Click Here To Show Diagram Code
[go]$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . O X . . . . . .
$$ | . . O X X . . . . . .
$$ | 9 3 O O X . . . . . .
$$ | 8 1 X X O O . . . . .
$$ | . 2 . X X O . 7 . . .
$$ | . . . . 6 4 5 . . . .
$$ | . . . . . . . . . . .
$$ ----------------------[/go]

is another endgame play and like before is too slow.
$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . 9 . . . . . . .
$$ | . . 0 O B . . . . . .
$$ | . . O B B . . . . . .
$$ | . O O O B . . . . . .
$$ | X O X X O O . . . . .
$$ | . X . X X O . O . . .
$$ | . . . . X X O . . . .
$$ | . . . . . . . . . . .
$$ ----------------------
- Click Here To Show Diagram Code
[go]$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . 9 . . . . . . .
$$ | . . 0 O B . . . . . .
$$ | . . O B B . . . . . .
$$ | . O O O B . . . . . .
$$ | X O X X O O . . . . .
$$ | . X . X X O . O . . .
$$ | . . . . X X O . . . .
$$ | . . . . . . . . . . .
$$ ----------------------[/go]

is a great point instead which gets [sl=PushingBattle]gets ahead in the pushing battle.[/sl] This move is simply huge - it builds fourth line territory, protects your cut, and has a large effect on the center. Black's

stones go from looking strong to looking very weak with this one move. Speaking of the cut, if black plays

, you can probably see how to capture that stone. (Hint:
Click to show).
$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . 0 . . . . . . .
$$ | . . . O X . . . . . .
$$ | . . O X 2 . . . . . .
$$ | . . 3 1 4 . . . . . .
$$ | . . 5 X 6 . . . . . .
$$ | . . . 7 8 . . . . . .
$$ | . . . 9 . . . . . . .
$$ | . . . . . . . . . . .
$$ ----------------------
- Click Here To Show Diagram Code
[go]$$W
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . 0 . . . . . . .
$$ | . . . O X . . . . . .
$$ | . . O X 2 . . . . . .
$$ | . . 3 1 4 . . . . . .
$$ | . . 5 X 6 . . . . . .
$$ | . . . 7 8 . . . . . .
$$ | . . . 9 . . . . . . .
$$ | . . . . . . . . . . .
$$ ----------------------[/go]
This diagram looks pretty realistic. I think black should play

next because the difference between white and black getting that point is huge in this variation as well.
As for the tiger's mouth variation, I think it should end up the same as the diagram above, just with

and

in swapped places.