I feel like people have no idea how it works and why it exists, so I'm going to clear it up.
In basketball you are allowed to take two steps in order to control your momentum when you are going for a layup. The two steps only count after you fully gather and pick up the ball. NOT WHEN THE DRIBBLE IS STILL LIVE. If this didn't exist your layup would get blocked all the time since you would have to completely stop before shooting the layup. When you are taking the two steps you are allowed to completely stop.
When you completely stop you are allowed to shoot the ball or pass the ball. In order to shoot the ball you naturally have to plant the your pivot which counts as a step. So because of this you can do a step through, since you already have to take a "step" anyways so might as well get yourself to the most ideal spot with that step.
Now the gather step exists because sometimes when you are about to pick up the ball, you are in an awkward position(one of your feet is still in the air), so you have no control over the ball. So the rules allow you to "gather" and gain control and plant your foot down before you start to count the steps.
In the Hugo Gonzalez clip, he stutters steps, which DOESN'T end his dribble so you don't count his steps. If you could only step once per dribble everyone would be playing at the same exact speed, which would make offense too easy to predict. He didn't pick up his dribble/ball kept rotating in his hand so his dribble didn't officially end.
When the ball stops moving/rotating that is when his dribble officially stops. Dribbling after the ball stops rotating/moving would be a double dribble or a carry. After the ball stops rotating, Hugo is in an awkward position(one of his feet is off the ground). The rules allow him to plant that foot before counting his steps(aka a gather). So after he does his gather, he takes two steps for the layup.