Many people have added notes, but here's my own little mini thing.
There are a couple of interesting thoughts concerning the hane at

, but first a quick note on your last diagram:
$$B 8 played at a or b depending on surrounding stones
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 O . . . . .
$$ | . a 1 3 4 6 c . .
$$ | . . . . 5 7 b . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$B 8 played at a or b depending on surrounding stones
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 O . . . . .
$$ | . a 1 3 4 6 c . .
$$ | . . . . 5 7 b . .
$$ | . . . . . . . . .
$$ --------------------[/go]
b is wrong. It should be at c. Here's the joseki:
$$B 3-3 invasion part 1
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . 0 . . . . . . .
$$ | . 9 2 O . . . . .
$$ | . . 1 3 4 6 8 . .
$$ | . . . . 5 7 . . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$B 3-3 invasion part 1
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . 0 . . . . . . .
$$ | . 9 2 O . . . . .
$$ | . . 1 3 4 6 8 . .
$$ | . . . . 5 7 . . .
$$ | . . . . . . . . .
$$ --------------------[/go]
$$Bm11 3-3 invasion part 2
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . O . . . . . . .
$$ | . X O O . . . . .
$$ | . 1 X X O O O . .
$$ | . . . . X X b . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$Bm11 3-3 invasion part 2
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . O . . . . . . .
$$ | . X O O . . . . .
$$ | . 1 X X O O O . .
$$ | . . . . X X b . .
$$ | . . . . . . . . .
$$ --------------------[/go]
To understand why playing b instead of

is correct, we should examine why it is necessary to play

. Suppose we omit

$$Bm11 3-3 white 12 omitted, variaton 1
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | 9 3 6 . . . . . .
$$ | 4 O 5 7 . . . . .
$$ | . X O O 8 . . . .
$$ | . 1 X X O O O . .
$$ | . . . . X X b . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$Bm11 3-3 white 12 omitted, variaton 1
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | 9 3 6 . . . . . .
$$ | 4 O 5 7 . . . . .
$$ | . X O O 8 . . . .
$$ | . 1 X X O O O . .
$$ | . . . . X X b . .
$$ | . . . . . . . . .
$$ --------------------[/go]
$$Bm11 3-3 white 12 omitted, variaton 2
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . 7 . . . . . . .
$$ | . 3 4 . . . . . .
$$ | 5 O 6 . . . . . .
$$ | . X O O . . . . .
$$ | . 1 X X O O O . .
$$ | . . . . X X b . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$Bm11 3-3 white 12 omitted, variaton 2
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . 7 . . . . . . .
$$ | . 3 4 . . . . . .
$$ | 5 O 6 . . . . . .
$$ | . X O O . . . . .
$$ | . 1 X X O O O . .
$$ | . . . . X X b . .
$$ | . . . . . . . . .
$$ --------------------[/go]
Without the defense of

, Black can clamp and escape. White will have to struggle to keep Black contained.
With that being said, why is this important for understanding

? Let's look.
$$B 3-3 invasion part 1
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . 0 . . . . . . .
$$ | . 9 2 O . . . . .
$$ | . . 1 3 4 6 . . .
$$ | . . . . 5 7 8 . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$B 3-3 invasion part 1
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . 0 . . . . . . .
$$ | . 9 2 O . . . . .
$$ | . . 1 3 4 6 . . .
$$ | . . . . 5 7 8 . .
$$ | . . . . . . . . .
$$ --------------------[/go]
$$Bm11 3-3 invasion part 2
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . O . . . . . . .
$$ | . X O O a . 7 . .
$$ | . 1 X X O O 5 6 .
$$ | . . . . X X O 3 .
$$ | . . . . . . 4 . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$Bm11 3-3 invasion part 2
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . O . . . . . . .
$$ | . X O O a . 7 . .
$$ | . 1 X X O O 5 6 .
$$ | . . . . X X O 3 .
$$ | . . . . . . 4 . .
$$ --------------------[/go]
What we find out is that the clamp can be used on this side with a similar result. Because we must play

to defend, we don't want to leave a weakness for Black to take advantage of on the other end of our chain of White stones.
Now, on to the next topic:
$$B 3-3 after Knight's approach, part 1
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . 8 0 . . . . . .
$$ | . 7 4 O 6 . . . .
$$ | . 9 3 5 a . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$B 3-3 after Knight's approach, part 1
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . 8 0 . . . . . .
$$ | . 7 4 O 6 . . . .
$$ | . 9 3 5 a . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------[/go]
$$Bm11 3-3 after Knight's approach, part 2
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . O O . . . 2 . .
$$ | . X O O W . . . .
$$ | . X X X a 1 3 . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$Bm11 3-3 after Knight's approach, part 2
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . O O . . . 2 . .
$$ | . X O O W . . . .
$$ | . X X X a 1 3 . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------[/go]
I won't go into all the details on this joseki, but this is one possible variation. The point in question is why

(also marked

) is not at a. You saw the result in your game.
$$B 3-3 after Knight's approach, tricked?
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . . . . . . . . .
$$ | . b 4 O a . . . .
$$ | . 7 3 5 6 . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$B 3-3 after Knight's approach, tricked?
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . . . . . . . . .
$$ | . b 4 O a . . . .
$$ | . 7 3 5 6 . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------[/go]
With

in place, Black uses

to strengthen his group before attacking at the cut at a. The idea is that White will play at b to try and keep Black surrounded. You did this in your game. You have another option ... play at a yourself.
$$B 3-3 after Knight's approach, maybe not tricked
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . . . . . . . . .
$$ | . 9 4 O 8 . . . .
$$ | . 7 3 5 6 . . W .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$B 3-3 after Knight's approach, maybe not tricked
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . . . . . . . . .
$$ | . 9 4 O 8 . . . .
$$ | . 7 3 5 6 . . W .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------[/go]
In the game, you had a stone already where

is in the above diagram (and your opponent had a stion where

is ). So, for you this sequence would end up with your sente, as your group is safe. Black's corner isn't really all that big, and the shape isn't great for him.
There are a couple more variations that are interesting. Suppose

above was one space higher?
$$B 3-3 after Knight's approach, Black forgot about surrounding stones, part 1
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . . . . . . . . .
$$ | . 8 4 O 9 . . W .
$$ | . 7 3 5 6 0 . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$B 3-3 after Knight's approach, Black forgot about surrounding stones, part 1
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . . . . . . . . .
$$ | . 8 4 O 9 . . W .
$$ | . 7 3 5 6 0 . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------[/go]
$$Bm11 3-3 after Knight's approach, Black forgot about surrounding stones, part 2
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . . . 1 2 4 6 . .
$$ | . O O O X 3 5 W .
$$ | . X X X O O 7 9 0
$$ | . . . . . . 8 . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$Bm11 3-3 after Knight's approach, Black forgot about surrounding stones, part 2
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . . . 1 2 4 6 . .
$$ | . O O O X 3 5 W .
$$ | . X X X O O 7 9 0
$$ | . . . . . . 8 . .
$$ | . . . . . . . . .
$$ --------------------[/go]

catches Black in a ladder ... disaster for Black!
One more variation, which might be useful to know:
$$B 3-3 after Knight's approach, different punishment, part 1
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . 8 0 . . . . . .
$$ | . 7 4 O . . . . .
$$ | . 9 3 5 6 . . W .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$B 3-3 after Knight's approach, different punishment, part 1
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . 2 . . . . . .
$$ | . . . . . . . . .
$$ | . . 1 . . . . . .
$$ | . 8 0 . . . . . .
$$ | . 7 4 O . . . . .
$$ | . 9 3 5 6 . . W .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ --------------------[/go]
$$Bm11 3-3 after Knight's approach, different punishment, part 2
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . @ @ . 5 . . . .
$$ | . X @ @ 3 4 . . .
$$ | . X X X O 1 . W .
$$ | . . . . 2 . . . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$Bm11 3-3 after Knight's approach, different punishment, part 2
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . @ @ . 5 . . . .
$$ | . X @ @ 3 4 . . .
$$ | . X X X O 1 . W .
$$ | . . . . 2 . . . .
$$ | . . . . . . . . .
$$ --------------------[/go]
Here, White has caused himself grief with his

stones in trouble. The better way to defend:
$$Bm11 3-3 after Knight's approach, different punishment, part 2
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . O O . . . . . .
$$ | . X O O 4 2 . . .
$$ | . X X X O 1 . W .
$$ | . . . . 3 5 . . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$Bm11 3-3 after Knight's approach, different punishment, part 2
$$ | . . B . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . O O . . . . . .
$$ | . X O O 4 2 . . .
$$ | . X X X O 1 . W .
$$ | . . . . 3 5 . . .
$$ | . . . . . . . . .
$$ --------------------[/go]
This is a bit subtle, but is highlighted when

is sitting there ... let's add

to the result of what's considered joseki
$$W 3-3 after Knight's approach, later reduction
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . O O . . . . . .
$$ | . X O O W . . . .
$$ | . X X X 2 X . W .
$$ | . . . 4 1 3 5 . .
$$ | . . . . . . . . .
$$ --------------------
- Click Here To Show Diagram Code
[go]$$W 3-3 after Knight's approach, later reduction
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O . . . . . .
$$ | . . . . . . . . .
$$ | . . X . . . . . .
$$ | . O O . . . . . .
$$ | . X O O W . . . .
$$ | . X X X 2 X . W .
$$ | . . . 4 1 3 5 . .
$$ | . . . . . . . . .
$$ --------------------[/go]
Now,

isn't supposed to be played immediately, but is something you have as a reduction later. In the previous diagram, Black has removed this weakness.
There are many, many variations of the 3-3 invasion. It's something that I think deserves some study from time to time for most DDK and even SDK players.