OK so my setup appears to be VERY similar to the diagram
here, except I don't have a second pump or a spray bar.
My tank has two places for aquarium water back in, though, and the one pump has a T-fitting on the output. Therefore, it looks like I can either (a) run the first line directly back to the tank, and the second line through the UV sterilizer, chiller, and back to the tank or (b) run the first line through the chiller to the tank, and the second line through the UV sterilizer to the tank.
I don't know which way is better, but I would think (b) would provide more even return flows on each side. I am pretty sure the previous owner had the UV sterilizer in-line with the chiller, though, so I don't know if there is an advantage to that.