Is it then the variable (or the variable accessor) that is impure, or the function that is impure?

The variable accessor is a function of type Variable -> Field, which the function sum is calling. Because of that, the function sum is impure, as it's calling an impure function. I would argue that the object is just a wrapper around a function, and in essence making the sum function implicitly higher order, as you're passing it a function to retrieve a value, rather than an actual value.

Math.random doesn't necessarily have side affects. If it's storing a seed value, and incrementing that each time; then yes it stores mutable state which is a side affect because future invocations of that function will then depend upon that pre-existing state.

If however Math.random is passively listening for background radiation through some sensor perhaps then it doesn't really have side affects does it? Nor does it have mutable state, for that matter.

But it would still be impure according the true definition of what is a pure function, I believe.

I thought it was a bit pedantic as well, at first. But looking at Haskell, I can see why the author made this point. Haskell functions won't accept random values as input, since they're monadic values.

Are there two identical inputs to Math.random that yield different results? If the answer to that question is yes, the function isn't pure.

Replace Math.random with this, then:

    ((j) => () => j++)(0)