Suddenly YouTube doesn't work for me any more. Nothing has changed since yesterday but today I get a daft message saying "Your browser does not currently recognize any of the video formats available". Well, so much the worse for YouTube! I never thought much of it anyway.

What I would like is an old-fashioned written explanation. I already looked in How Stuff Works but couldn't find anything relevant there. Is the plastic skin of these toys anything like a memory foam?

The shell is probably a bit like that, something that starts out 'undefined' until it sets in a form, where the chemical properties change into a more persistent, rubbery state.
The filling is probably done because the material the shell is made out of would be too rigid for a full cast of it, so it's a balance between its desire to come back into its shape and having some loose liquid thing to give it the heft needed to behave the way it does.

The liquid filler contains a strong surfactant. There have been some nasty stories about children who squeezed the toy too tightly, and the skin broke and shot a jet of the stuff right into the child's eye. Apparently it causes very severe pain and could cause permanent damage.

I've noticed that these things are sold on the street, which probably means they are cheapos from China.

Sounds about right, and it's proably quite enticing to squeeze it, too.
Entices the child to squeeze it because the goo inside strains against the rubber, but not designed to be a toy of that kind, like a stress ball.

I remember those snot like things, too, they collected lint like crazy, but at the very least they were their own whole package.
They should just bring those back, but then again, 'slime' had some chemical in it, too (not that the 'snot' stuff was slimy per se). So who knows.
Damn chemicals. Oobleck balloons would be a solution, if its properties were inverted...but, alas.

This is the brief text of that youtube page I posted

"Are your splat balls safe for children?

Yes, our splats contain only filtered water so they’re the safest available. They’ve been tested, meet US regulations, and are perfectly safe. Most company’s splats aren’t tested and contain lead and phthalate levels that do no comply to US toy standards and regulations (it’s illegal for them to sell them, but it is so easy to get them into the country since no one is policing it). We continually change our splat recipe to meet or exceed changing safety standards. We spend 30% more than our competitors to make our splats so they are safe for your children. The water in our splats, unlike most of our competitors, is safe enough to drink (I don’t suggest it, though"

https://www.splatback.com/faqs/

If it only contains filtered water then how do they change the recipe? Maybe the outside is a recipe?

They changed the recipe by using filtered water! Like I said, most of these things contain detergent solution. But that in itself suggests that detergent/surfactant is needed for the full effect. As we've seen in this thread, there are splat toys that stick to surfaces but don't deform very much.