'Some' are correct, O'Hart's Irish Pedigrees is not a very reliable source. But to determine where ancestors of those of a surname 'Coulter' may have lived, you'll need to research back, person for person. That is, to identify the parents, successively, of each generation going back in this line (if that is even possible). Hereditary surnames were not in use that far back and spelling varies so much so that trying to match names is very unreliable. Also, what few records survive from this early period typically mention only a very few, and usually very prominent individuals.
There was always some movement of individuals and groups between Scotland and Ireland during the medieval period. At times larger group movements took place. But a migration of any group between the two locations, at any particular time in history, does not necessarily mean the person's you might be researching were a part of that movement. Not without some proof that this person went to this place (and if you're lucky, for what reason). Still, you won't know if you don't look into it. Best of luck in your search.