I just snapped these two pictures. I guess the question is answered though, and we've got a wild cherry. The good news is, maybe we can get around to making a little jelly. The bad news is, we'll probably be buried alive in dye-rich bird poo now that it decided to be real fruitful. Had I had any idea what we had here, I probably would have cut it when it sprouted. It has already trashed spots on the shed roof from branchs hanging on it before I got 'em pruned; the berries aren't going to help any.
I just popped one ripe cherry. It will take a few to add up to much flavor, and maybe not much then, at least without a little sweetning to bring it out.

So, I guess there was a reason I had never heard of birch berries.