On some websites it’s also considered an Irish surname as many Coulters settled in Counties Antrim and Down.
Exactly my point. The name and clan came from Scotland. Scottish Coulters may have settled in Ulster; but the name is Scottish; and Ulster has been since the time of the plantation like a different country, certainly with a different religion and allegiance to England, than Catholic Ireland, which contains the descendents of the aboriginal Irish peoples. Anne is a protestant and it is unlikely that her DNA goes back to the native Irish, but to the Scottish Coulters on her father's side.