Any communicated argument will...involve both reasoning and rhetoric; but the two cannot be simply conflated. You can use the same tactic of reasoning with a very different tactic of rhetoric, and vice versa. So we need to distinguish the two.
Fallacies like ad hominem, ad misericordiam, etc. turn out to be various rhetorical formats of the one fallacy of ignoratio elenchi. But what is the rational fallacy in straw man? It would appear at first glance that it is also a form of ignoratio elenchi: it leaves the other person's position unrefuted by presenting a different position (a weaker one) than it. But if this is so, not every simplification, or even obvious oversimplification, of an opponent's arguments would be a straw man. It would only be those cases where the oversimplification has so changed the nature of the position that a refutation of that position can have no bearing on, is for all practical purposes irrelevant to, the original position. Thus to show that something is a straw man it is not enough to note that it oversimplifies, or even misrepresents through its oversimplification, the original position; it has to be shown that the misrepresentation is serious enough that knocking it down doesn't introduce anything that tells against the original position.
Thus it seems to me that Raley and Talisse keep confusing the rhetorical and the rational aspects of the informal fallacy in their essay; they identify a rhetorical tactic, simplification of the opposing position, but incorrectly assume that the tactic itself is fallacious. (A good example is the Bill Clinton example: no doubt this does simplify Dole's position, but the point would seem to be not refutation of Dole's position, but a proposed contrast of the general sort of position Clinton wished to support with what he saw, or wished people to see, as the general sort of position Dole wished to support.) In reality, since straw man fallacy is a fallacy of irrelevance, this rhetorical tactic isn't shown to be a straw man unless we have shown it to introduce irrelevance.
As for the "twist" on the straw man that they talk about, the "weak man fallacy," it is, as far as I can see, no fallacy at all: it is simply the argumentative strategy of attacking weak arguments. Raley and Talisse's examples do a fair enough job of showing why this is often a faulty argumentative strategy; but fallacy and bad or even misleading strategy are different things.