The name ‘Ireland’ is a geographical rather than a political description. It refers to the entire island – i.e. the 32 counties. If Ginge’s original question had asked ‘born in the Republic of Ireland’ than he would be correct. McCoy was born in Ireland, as was Dunwoody. Even Ian Paisley was once quoted that he ‘would never repudiate the fact that I am an Irish man’. Whether they consider themselves ‘British’ or not is their own business. Since the Good Friday agreement, anyone born in Northern Ireland can hold a ‘Pas Éireannach’ (an Irish passport) if they so wish.
In racing terms there is no border. Both Down Royal and Downpatrick are in the UK but they do not fall under the BHA’s control. Similarly any trainer from the 6 counties or any horse bred there will have the initials (IRE) after their name on a racecard.