I finally got a good backup and a good verify.
Your suspicion about eSATA devices led me to try changing the drive over to a FireWire400 interface, which it is capable of doing. (My ASUS mobo doesn't have a FireWire800.) Something I did initially produced two partitions on that one drive, the second was a small 2GB piece. Needless to say, I reformatted the drive. The size was larger now, so obviously the smaller second chunk before reformatting couldn't be seen. I copied a backup I made to a USB disk, which verified successfully, onto the reformatted drive and it verified there. Next, I did a backup to the reformatted drive, which was successful. Then I verified it, which was successful, status "Completed."
Next, I'm going to go back to eSATA and backup/verify. Since I discovered the 2GB piece and then reformatted, it might work now.
Remember, my "real" problem was that I ran backup successfully, but the verify failed. (At first, I didn't scroll down to see the errors.) I suspect that the original format terminated abruptly though it was essentially completed, thus the completed/failed confusion.
So now, thanks to your advice, I will proceed to buy your product. I am convinced that not only is it good, but that the support you give is second to none.
I will advise when I try the eSATA again to see if it is successful because of the reformatting.