But that's A-theory, not presentism, which is being explained, right? This paper <pdf> claims there's a distinction

Yes. One can certainly be an A-theorist without being a presentist. Some people really have subscribed to so-called "moving spotlight" theories. (Hermann Weyl was an example.)

I'm less convinced that anyone was ever a presentist but not an A-theorist. The paper you cite doesn't convince me for at least the following reasons.

First, the paper doesn't even argue that any non-A-theorist presentists have ever actually existed. Rather, the paper attempts to show that such a theory is, as it were, technically possible.

Second, I don't buy that the paper succeeds even at this. The author constructs the theory in Section 4. But the constructions essentially depends on a loophole: A-theories must posit A-properties, he says, but existence is not a property. Then, in Section 5.3, he deals with what seems to me to be the obvious reply. He allows that maybe A-theories only require A-facts, and not necessarily A-properties. If existence is a fact, then his construction fails. His reply is that "it is still possible to be a presentist without being an A-theorist: we need simply deny the existence of facts. ... If there are no facts at all then there are no existence facts. ... This is not an unreasonable view. There are metaphysical systems that do not posit facts—versions of substance theory, bundle theory, and so on."

I find this unconvincing. I don't know enough about these other theories to know how they get by without facts. But I suspect that they introduce some kind of things, call them faks, that do the work of facts. I suspect that the A-theory could just as well be held to require only that there are A-faks.