I think I grasp now that doing multiple integrals over a Gaussian plus some extra polynomial junk in the integral corresponds to having different types of edges, and an "extra junk" term like x^2yz means: now you are allowed a vertex that wants to join to two x-edges, one y-edge, and one z-edge. So I guess a goal for myself now is to stare hard enough at the QED Lagrangian and see if I can find out where it's saying "one photon, one outgoing electron, one incoming electron". There's a lot of spin algebra going on that's occluding my view at the moment, though.
(The astute reader may notice that I erroneously wrote A, B, and C in the diagram where I should have written 1/2A, 1/2B, 1/2C)
