Is the rinsing needed, strictly speaking? Is there something special about spitting the water out afterwards instead of e.g. swallowing it?

Who said they spat it out? Rinse doesn’t necessarily imply spitting something out. You spit out mouth wash after rinsing because you can’t drink mouth wash. But sugary water is something you can drink.

Yeah I don't necessarily spit it out. It's getting it off of your teeth that is the important thing.

Your stomach is already full of whatever it is you just ate. 0.1% more from whatever remains in your mouth is not going to hurt it.

You should rinse and spit it out in the morning, don't swallow as you will be swallowing significantly more plaque from the over night build up. After initial morning cleaning swallowing should be fine.

Why would swallowing the plaque be significant?

It destroys your heart, possibly. Plaque builds up in arteries, etc. I just don't know if dietary plaque gets into the bloodstream. Don't think there are many studies. But plaque left on the teeth is known to cause heart problems and be lethal so I imagine trying to keep it out of our systems has to be better than not.