Wait...I may be an idiot...
I went through this whole rigamarole of downloading the "Install OS X Lion" app onto a second mac, then copying that App over to the "Applications" folder of the old mac I'm trying to update. Then, when I looked in the "Applications" folder of the old Mac, sitting there right next to "Install OS X Lion" was an App called "Install OS X Mavericks". I have no memory of downloading that particular app, but perhaps it did, and perhaps once it did, the system prevented me from downloading the Lion installer...which would make some sense.
As of this moment, I am upgrading the machine to Mavericks, which seems like a win all around.