After about half an hour, I got a call from an AAA dispatcher saying, "The driver is at your car and can't find you."
We started walking back out to the van and explained that we'd only been doing what the original call operator told us to do. The dispatcher said that he didn't know anything about that. Neither did the driver, when we told him the instructions. Sigh. I reckon that if we'd stayed with the van, he would have gone to the passenger pickup area instead.
Anyway, jump-starting the van took very little time, and we were on my way. I took the van out onto the freeway and ran up to Whipple Road and back -- about a 15-minute round-trip -- in order to pump some more power back into the battery. I hope it's enough to start it tomorrow, and I also hope that these two run-downs in two months haven't ruined the battery, which is only a couple of years old.