"British" is the inclusive term of the: Northern Irish, Scots, Welsh, & English as I understand it. As I understand it, the 'Union' Jack represents all of these, hence the
'United' Kingdom of Great Britain and Northern Ireland, as the proper name for the sovereign state. Ireland is a completely different proper country. The Irish flag, being a completely other country, has nothing to do with them except that they are adjacent nations. Its like saying you are a United States America who happens to be from North Carolina, so calling somebody from N. Carolina an American is like calling a English man (Or Scot, Welsh, & N. Irishman) a Britain/British. Now your example, calling a Scot an Irishman is a bit like calling a Carolinian a Mexican. Completely different country in the proper sense. Did I get this right you
Euros?
Btw, your probably confusing Scots with Leprechauns, who are Irish, a common mistake given their accent.