Bettye
What I found yesterday in Burton Fleming records were Robert Cowtass, son of Robert, bpt. 1779; Ann Cowtass, bpt. 1782; Jane Couton, bpt. 1785; James Cowton, bpt. 1787; Richard Cowton, bpt. 1790; Isabella Cowton bpt. 1792. If the father is your Robert, that would suggest the family then moved to Thwing, where Vickerman and Richard were born. (I can't see a burial for the previous Richard).
I looked back to see if I could spot a marriage in Burton records which might clarify whether there were actually two families, but the quality of the records is very variable and I couldn't see anything. I did wonder whether the Robert who married at Brid in 1778 might have given his home parish simply as Burton. According to my reference book, banns were supposed to be called in both parishes from 1823, "but sometimes earlier." So it's quite possible that didn't happen and the vicar assumed it was Burton Agnes.
Just came across something, but perhaps you already know. It seems there is a will of Robert Coultas of Thwing, proved in April 1821. You can order it from the Borthwick Institute at York (
http://www.york.ac.uk/borthwick/remote-services/copying/order-form/), pay for it online and they will send you a scan, which you need to download within a deadline. It's not expensive and they are very quick. There's no saying how helpful it will be, but it's worth a few dollars.
When I can - I'm going to be away - I'll have a look in the archives to see if there are any more clues.
All the best
Joyce