Rick, you will find this spelling (Coulter) to be correct, he signs his name as such in his military service records. In addition to his MDAH survey, E. D. Coulter also applied and was approved for a Confederate pension in Arkansas. If you need these pension records, they may be procured here. http://history-sites.com/research/index.html
Arkansas Confederate Pensions
Surname: Coulter
Given Name: E. D.
Application Number: 2948
Widow Surname:
Widow Given Name:
Company: I
Regiment: 35
State Served From: Mississippi
Division: Infantry
Pension County: Ashley
Death Date:
Application Year: 1904
Widow Death Date:
Comments: approved
Family researchers indicate his full name was Edward Dixon or Dixion Coulter , he and his wife, Nancy Elizabeth Barmore, are buried here.
http://www.ashleycountyledger.com/history/cemeteries/cleveland.txt
These same family researchers have also provided a photo (post-war) of Edward & Nancy Coulter, I have a copy if you are interested.