I think the suggestion that it was indeed just the slide from stopping a run is correct. I'm not sure how well everyone understands these movement states in Melee and Brawl so I feel like I should explain what they are and then add my speculation on what may be changed here.
Slamming the stick to the left or right causes your character to go into a fixed burst of speed in the respective direction. This is a "dash". It has a fixed speed and distance, and if you continue to hold the direction as it completes, you will seamlessly transition into a "run" state which is just a continuous state of moving forward. This distinction should be well known to anyone familiar with Melee Fox; his dash was a lot faster than his run so it was better to repeatedly dash instead of allow a continuous run to occur ("foxtrotting" was the community's name for this tactic). Notable about a dash is that it has to complete before you can just stop; Melee let you interrupt it with another dash (Brawl did not), but you couldn't just dash and then stop mid-dash without doing something else. Releasing the stick while dashing results in the dash completing and then skidding to a stop whereas releasing the stick while running immediately transitions into skidding to a stop based on your current speed and friction. Since dashes are generally faster than runs, releasing the stick during a dash will usually result in a longer and further slide, but that exact dynamic is character specific.
My guess for what is changed is what that the slide is now a totally free action in which the player can do whatever. Before it was treated the same way as dashing or running wih the lock to dash attack as a standard attack though the player could jump or shield to open a few more options (most notably up smashes and up specials). If true, this is a really big deal, but it's not the same thing as wavedashing at all. With wavedashing, it's about the ability to instantaneously get momentum in which there's zero commitment. Here, you're still going to have to finish dashes so if you use it with a dash it's basically what wavedash would be if it had start-up, and it's unclear how momentum is carrying here (Mario seems to stop completely with the jab, but Mario's friction isn't exactly bad so he may have just been done moving). More critically, you can run across the stage to a distant opponent and then attack freely more quickly; since you can stop a run whenever, this could allow for those with a bit of tech skill to basically do whatever they want out of running which would be a really big deal.
That's just my speculation; it could change before the final game, or I could be reading this wrong. Either way, it was a pretty fantastic catch to notice something wasn't normal there; that is so early in the trailer that it's really hard to notice, and the Fox situation is so chaotic that it's hard to even see what he's doing. This would be a thoughtful tweak of base mechanics which is something we've already seen Sakurai do twice now (with the ledge mechanics and the rapid jabs) so i wouldn't really be surprised if this or something like this really were in the final game. I continue to really look forward to having this game in my hands and being able to see for myself exactly how everything works.