The cut  

What happens next will depend on the surrounding positions. Even if white does end up sacrificing the original 1 stone black will crawl on the 2nd line a bit so white will get stones in this area in sente and thus be able to use them to sabaki and reduce black's potential here.

I note that in the pro game example you posted black already had many stones in this area so black judged he could handle the fight.

The other cut

No other cut, part 1  

The other cut in this position is also fixed by defending the 3rd line cut with a hanging connection .

Yes other cut, part 1  

However, if black doesn't defend, then white could play something like this. (Note if black has the ladder to the lower right he can just capture in a ladder with 4 at b, but nets are better than ladders as they are local and don't give a ladder breaker). If black pulls out at a white may extend, black crawls on the 2nd line a bit and maybe white can hane and kill. Or white can answer a at c, if black answers at d then white can e to capture. If black omits d to save his 2nd line stones then white can escape at b which cuts black and a fight develops, the circled black stones are now cut and weak.

2) Eyeshape

Eyes  

a Is almost certainly an eye, b can easily become one.

3) Thickness

Attacking with thickness  

There are many ways the wall could be used for attacking, in this diagram white has invaded and black can pincer at to push white into his wall. If he hadn't made the hanging connection then after the pincer at white could cut at a and now the fights from the previous diagrams are really easy for white with the support of . So if black answers to make the hanging connection, white has sente to extend and make a base at b, which also threatens to slide to c.

4) Endgame

Defend: white gote endgame  

Even ignoring all the cuts, the handing connection is good for endgame (and better than the solid connection, it also makes better eyeshape than the solid) as it makes white's hane connect there gote.

Tenuki: white sente endgame  

If you tenuki white gets this endgame in sente and you have 2 fewer points.

Tenuki: clamp  

In fact white has something rather better than the 1st line hane in the last diagram, though it can be gote: the clamp tesuji of 1. Black can't descend to a or white cuts at b and either the circled on squared stones are captured.

Tenuki: clamp, variation 1  

Black can play this way and take sente, but he has lost many points on the top side. a is a big point, black may actually exchange it for b before tenuki because if white plays there he then has to start worrying about eyes for his wall.

Tenuki: clamp, variation 2  

In this variation black plays 4 to keep some territory on the top. However, he takes gote to do so which means white's clamp reduced in sente which is good for white. Here white can either connect from the atari of in sente and black defends the cut of a, or white may cut at a and fight a big ko.

5) Honte

Playing honte to not leave bad aji is one of those things that takes time to appreciate. The aji of the cut gives white also sorts of tactics, here is one idea:

Bad aji  

Black is trying to make some points at the top and omit defending the cut. White contacts at . Now if black stands at b the white cut at a is severe as will be useful in the fight. If black answers by defending the cut, white gets to play 2 moves in a row on the side stone with b.

Aji in real game

Artem92 8d (White) vs duelist 6d (Black) on KGS  

Here is an example from a real game, http://files.gokgs.com/games/2010/8/15/artem92-duelist.sgf, of the bad aji from leaving the cut. Though in this game black was right to not defend the cut as it would be overconcentrated and sente was very important. White would like to enter the right side to stop black getting too much there. Crawling on the 2nd line at a is not so good, so artem uses the cut aji.

11-20  

White has now entered the right side, alive as connected to the corner, on the 3rd line, in sente! However, I think black made a mistake and 2 should be at 3 to avoid the squeeze.