Did you remove the pinion flange? By the looks of this 2nd video you need to tighten up the pinion nut. I don't think there should be that much movement on the pinion. There is a special "crush sleeve" inside, that is supposed to be torqued to a certain reading, or else there is a technique I have used in counting the exposed threads on the shaft.
If you have this much play I don't think you will drive far without doing some damage to the mating gears inside the diff.