This doesn't (necessarily) have anything to do with rotation speed, and the article doesn't say it does. The fact that Betelgeuse is pulsating more quickly does not imply that it is rotating more quickly.

Making an entire star suddenly spin twice as fast on its axis would require a staggeringly, absurdly huge transfer of energy and angular momentum. The mass ejected from Betelgeuse was much, much smaller than the star's total mass, and would not have been capable of causing such an effect.

In that example, your arms are still rotating about an axis. The moment of inertia of the rotating body increases, so the angular velocity decreases. Imagine instead that the ejected material is no longer rotating about an axis (say, you detached your arms and flung them away in a straight line); the moment of inertia of the remaining rotating body _decreases_, so the angular velocity increases.

Imagine a figure skating spinning fast. Then holds out her arms and slows down. Then someone chops her arms off.

Sort of like that.

Imagine instead that you’re sitting on a swivel chair and you fire a gun with your right hand. This is more akin to how ejection of materials work. When you put your hands out on the swivel chair, the hands are still attached and are forced into orbit around your torso, when you eject material this isn’t the case, and it leaves the system.

Your arms are still attached, unlike ejected mass.

I don't think the pulsations are due to rotation. It grows and shrinks periodically, causing changes in absolute magnitude; and there seems to be more than one period. The various periods seem to be related harmonically.