Really interesting! Looks like an odd inversion of the side-step in the picture, but its doesn't sound so in the description. Definitely a bare knuckle variant (obviously).
That cross step showed up in instruction books for a long time; it is in the Haislet book, which is the Bible of boxing instruction books. When the fighters have their lead feet close, it works well, especially to get off the ropes and out of corners. The current trend has fighters more squarely in front of each other, the lead foot towards the center of the opponent.
I've seen fighters use the cross step, and I've seen some gyms practice it. The ones I've seen did it with more knee bend and immediately pivoted on their left foot after the cross step. There is risk with everything and IMO, the risk with the cross step is that while stepping with the left across the right, the fighter is off balance and any punch that lands at the time could cause an "off balance" knockdown. A two point round is a two point round whether the fighter is hurt or not. Another problem we found is that some fighters will naturally slide their left foot as the other fighter cross steps causing the fighter doing the cross step to either trip, fall, or have to grab the rope to keep his balance. We liked it after seeing some fighters use it an amateur tournament (no two point rounds there!) and we experimented with the cross step. After trying it, our guys thought it was easier to just move their right foot first and they could still get the angle with less risk. But, they know how to use it now.
Wouldn’t it also make you highly vulnerable to right hooks (at least against an opponent who was trained to recognize the maneuver)? Looks like it would be a good athleticism-based move though, if you have faster reflexes and feet than your opponent.
IMO, the cross step is like everything, there are trade offs. The illustration shows the fighter doing the cross step without much knee bend, the guys I've seen use the cross step, bend more and then pivot as soon their left foot sets after the cross step. Of course if you knew the opponent did the cross step in certain situations, and you studied video you could counter it by pivoting and using the right, or by just bumping him while his feet are crossed, or many other ways. As you mentioned, if you're a quicker athlete you can get by with things others can't.
It's more of a draw move, in that it works well with an opponent that you've got really stepping with his jab. If you are moving as he is jabbing, you're around his left shoulder and there is no way he can get to you with a right hand, or any other punch, without picking up his feet and turning around. In my mind that is the objective of the move, to make him turn and hit him with a left hook as he does it. You can make him turn into a right hand as well. You need to know where your weight goes when you step and again as you pivot immediately after the step. Incidentally, in the Haislet book he uses this move as a way to land a left uppercut or hook to the body. He describes using an outside parry, cross step, body punch, then pivoting out. Very subtle transferences of body weight there but, once you get the feel of it, a very effective move. Not an 'athleticism' move either. I was in my late 20s when I read about it, worked it out on the heavy bag, and used it in sparring right away.
