The differential yoke nut is tightened to a torque spec. Probably 150 ft lbs. Check that spec. The rotational "play" you are showing has nothing to do the the tightening of that nut. You are feeling the pinion/ring gear backlash. The tightening of that not can have an affect on the axial end play/preload of the pinion gear if the backlash is previously set. Tighten it tight and you are done.
You may want to wait for other answers before tightening to 150.

Also search "phasing" of a drive shaft to be sure you install it correctly.

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