This is an update, not a real solution.
Well, my original question is sort of academic now, because if I use the USB interface on my external drive adapter, I can find and format the device under the Disk Management tab. Unfortunately, when I change back to the eSATA connection, the drive is still not recognized, even though another (old) Seagate drive works fine on that interface, in the same slot. I guess I'll just fall beck to the USB interface for my backups, for now.