A few years ago I was working Five-Star and got involved in a discussion regarding footwork off the catch and off the dribble. For a few years I went back and forth between the 1-2 step (stride stop) and slide stop until about a year ago I decided that players need to know both.
Here is why: The quickest way to stop off the dribble is a stride stop. Almost always the quickest way to get squared to the basket off the catch is by using a slide stop. While I do agree that a 1-2 step is quicker than a jump stop, a slide stop is not a jump stop. It is a short, low to the ground slide to square your feet. The best term to describe this concept is "square in the air".
The argument I heard for 20 years was that "you should use a 1-2 step so that your feet are the same every time." Well what about the situation where you need to get a shot off and the quickest way is "square in the air"? Do you teach your kids to play comfortable or to be as quick as possible without hurrying? You need to train your feet to play in specific situations.
So, to summarize:
Quickest way to get perfect feet off the dribble: Stride Stop (1-2 step off the dribble)
Ideal Situation: You are driving hard to the bucket with the defender on your hip or out of the triple threat.
Quickest Way to get perfect feet off the catch: Slide Stop (A short, low-to-the-ground slide to square to get "10 Toes to the Rim")
Ideal Situations: Coming off a stagger or anytime you need to square to the rim. Also, off a post kick-out, dribble fill, middle wrap or baseline drift.
Please remember that a slide stop is not the traditional jump stop. If you are jumping too far off the floor to "square in the air" it will be slower than the 1-2 step.