Each conducting material has its own mix of charge carriers, ranging from mostly electrons to mostly holes, or from mainly anions to mainly cations. This isn't really one of those situations in physics where you can change the description without changing the basic physics. The sign of the Hall effect, for example, depends on what type of carriers are present.
So let's proceed to follow Purcell's discussion in which the electrons are the sole charge carriers. I'm out of town at the moment, and thus don't have my well-worn copy at hand, but will work from your description. From the point of view of those electrons, yes, they're in a net-positively-charged stretch of wire, just as you argue. However, the other side of the current loop has electrons going the opposite way and looks net negatively charged to them. Overall, the current loop looks neutral in each frame, although the distribution of charges within it is frame-dependent.
But wait, you say. If the electrons see a local positive charge density, doesn't that mean there's a force attracting them to the wire? Yes. So what about in the lab frame? In that frame, the current density makes a magnetic field looping around the wire. The electron velocity crossed with that field makes a magnetic force attracting the electrons to the wire. So the basic physical effects are the same in either frame, although the name of the force changes.
(published on 05/01/12)