Am i right in saying, that when the button is pushed the LED should remain on all the time the button is pushed? and then as soon as you release the button, it should turn the LED off.
Sounds exactly right to me. The Button example clearly only will illuminate the LED when the button reads HIGH.
The Debounce example has misleading comments - the comments imply the LED only changes when the state goes LOW to HIGH when in reality any time the state changes from the previous (after 10 millis anyways) the LED will change state as well.
So in effect the Debounce example is just a different way of doing the same thing.
I hope this helps,