Left-right play at bottom bracket

This is actually an easy fix, but requires some special tools.

On Shimano cranks, the left hand crank arm clamps the axle. When this is installed the crank and axle are pulled together by tightening the plastic cap that threads into the axle through the crank arm. This sets the bearing pre-load and removes any axial play.

From the video it looks like you have lost the cap and that has allowed your crank to slip off the axle a little resulting in that axial movement.

Once you have a replacement cap. all you need to do is loosen the crank pinch bolts, finger tighten the cap to set the pre-load (the cap should not be over tightened) and re-tighten in pinch bolts.

However, you need a special tool such as the Park Tool BBT 1 -2 to tighten the cap. Also you really need a torque wrench to tighten the pinch bolts properly to avoid over-torquing them and stripping the threads. Conversely, under tightening the pinch bolts allows the crank arm to walk around on the steel axle which deforms the splines in the arm. Both are rather emotionally traumatic experiences as you realize you now need to buy a whole new crankset.

This is a good video showing how the crank is removed and re-installed, including setting the bearing pre-load.