Yes. It is the same Jesse Frank Thompson. I've not been able to figure out where people have gotten the "Franklin" name yet.
I've run across the "Delmar" thing before. It comes from this:http://i18.photobucket.com/albums/b103/cobweb2/Pict0120.jpg
(Sorry, I can't seem to find the title/author on this computer). I've not found any record of a Delmar Capps in the family. I have also found that based on Census records, William Thompson's father was most likely named Josiah.... not Joseph.