In the United States, they tended to identify as "Scots-Irish", and had a much more established place in American society (for example, at least three Presidents that I know of were Scots-Irish in the 19th century).
Scots-Irish isn't quite the same thing. Those referred to as 'Scots-Irish' in the US are descendants of Scottish (largely) Presbyterian planters. People who adhere to Anglicanism generally wouldn't consider themselves or be considered 'Scots-Irish'.
About half of Irish (whole-Island) Protestants today are Presbyterians concentrated in the Ulster Plantation (Northern Ireland); those are absolutely the same ethnic group as the Scots-Irish, despite that label not being used in the UK.
