Wait wait wait. I've just noticed something. The tutorial only talks about adding one material to the new mesh, the one with the grass texture, having deleted both the others. But whereas my material box says "1 Mat 1," the one in the screenshot says "2 Mat 2." Am I a material short that would make this work?
EDIT: this is what comes of thinking I can fix this myself. I tried pressing the "UseAlpha" button on the grass texture, because I thought, you know, the vertex painting is supposed to be an alpha thing, right? Didn't work, of course, and now my soil has sharp edges again, and nothing I can do will bring back the shading.
Would anyone consider letting me send them the age files so that they can see what I've mucked up and tell me?