"Heres one to play with. Richard Gleave. Waltz. First three of a Natural. Drive on one. Swing on two. Sway at the end of three."
Except of course that what your are quoting is only a very rough guideline.
Some sway will have develped by the end of step one. Not as much as will have occured later, but it will already have begun.
And the actual swing - that begins in step 1 too, because the upswing is nothing more than the natural followthrough from the downswing of the previous step 3.
But you associated the swing with step two because that's where it's most obvious, and the sway with the end of step 2 and start of step 3 because that is where the sway is most obvious.
It's just that were it's most obvious is not the only place where it occurs.
And why doesn't the book mention sway on step 1? Because it only comes in just before the end. Similarly, it mentions sway on step 3, even though in fact it's likely to be gone before the very end.
An approximation has to leave out some of the details - but if you don't want to look like a robot, those details left out of the approximation had better be present in your dancing!