Which force is doing the work here?

Let's make a simple example. A block with a compressed spring attached to it is on a frictionless horizontal surface against a stationary, immovable wall. The spring is released, and the block is then pushed away from the wall, thus gaining kinetic energy.

The relevant forces here are 1) the force between the spring and the block and 2) the force between the spring and the wall. Which force does work here? Force 1 did, because it is applied over a distance. The energy is transferred from the potential energy stored in the spring to the kinetic energy of the block.

In your example, the skater is the block, and the arms/muscles are the spring.

Biological systems are really tricky in physics class. Our intuition of how they work rarely lines up with how they actually do.

In this case, the answer is almost correct. Cleonis is correct that there is some work due to pushing the earth, but it's negligable. We can really say that their answer is correct unless we're talking to a rules lawyer or are solving the riddle of the Sphinx.

The trick is that you have to break the skater apart into their component bits. Consider just her hand. It doesn't move, so clearly no work was done. Now move up to her forearm. There's some movement here, so we can see that probably some work was done to the forearm.

Move up futher to the upper arm, and you see much more movement. And finally, when you consider the core of the skater, she's clearly being moved a lot. Work is clearly done on the skater.

If you were to look at just her bones, muscles and tendons, what you would see is that the bones of the hand are stationary, but the muscles and tendons pull on the bones further up her arm. It is this interaction where work is being done, putting lots of force into a movement over a distance (or, alternatively, a torque into a level arm, which you will learn is an equivalent way to think about it).

And in theory, a teeny-weenie bit of work is being done by her hand on the railing to move the Earth further away from her. (or more practically, she will bend the railing ever so imperceptibly)

I think I agree there is something off in that quoted section.

The purpose of that section is not practical application, the purpose is to bring an abstract concept into focus. And that means the usual simplifications for practical purposes should not be used.

There is a physics joke that goes as follows: When Arnold Schwarzenegger does his push-ups, he is actually pushing the Earth away from himself.

In the case of the skater the standard for-practical-purposes simplification is: she is in effect pushing off against the entire Earth, so we treat the railing she is pushing off against as immovable.

But: that simplification violates the third law. It is more instructive to think of a setup where the third law is visibly involved. You can have the skater push off against a sled that is, say, twice her mass. And when you have a clear concept of the third-law-visibly-involved case you can take that to the limit of pushing off against something that is vastly more massive than the skater.