I realise that in many cases it is a matter of probability rather than certainty. I can't trace my line back to Ceallachain of Cashel, for example, but I can trace them back to Cork, and it is an unambiguously Gaelic name. Others may not be so fortunate as to get that far.

The issue of Scottish v Irish is more complex than it appears.

AFAIK, a group of Gaels from the Dal Riada tribe, from Ulster, Ireland, settled in Argyll, Scotland around the 10th century, about 100 years before surnames appeared in Ireland, and surnames only appeared in Scotland even later than that. In both places most surnames came from given names, so you have largely unrelated clans on either side of the Irish sea that have the same name only because their founders had the same given name. The Irish McGraths and Scottish McRaes originate in the same given name, but are not directly related. A more familiar example is Kennedy, where the Scots clan is separate from at least two separate Irish clans, all founded by men whose given name was Cenneidhe.

Then you have the plantation about 500 years later, where mostly lowland Scots moved to Ulster, but they did include a proportion of highlanders whose ancestors may have came from Ireland in the first place. I know a feller who is an Ulster protestant who has a scroll proclaiming his surname to have originated in Argyll, exactly where the Gaels first landed in Scotland from Ireland hundreds of years before his family went from Scotland to Ireland!

If that weren't enough, you have clusters in Glasgow and other places in Scotland where there are people of more recent Irish catholic origin, who came to Scotland from Ireland around the 18th and 19th centuries. In fact, there is one guy who may be distantly related to me who claims our family came to Cork, Ireland from Glasgow, Scotland. Even if true, it is a Cork name, so if they did that they must have gone to Glasgow from Cork first! Moreover, he has them living in Cork in the early 18th century, which is perhaps a little early for them to have emigrated from Cork to Glasgow and then back again to where they came from, but nothing is impossible.

OTOH, someone who assumes MacCarthy to be Scottish because it begins with Mac has not done much homework. Maybe she was just beginning to figure out where they came from. Actually, the MacCarthy clan and the O'Callaghan clan are both descended from Ceallachain of Cashel, along with the McGraths, the Sullivans and one Kennedy line (but not the one leading to JFK, that one comes from the Dalcassian tribe, and all the above are Eugenians).

However, it is believed that some took the name upon joining the clan, which makes all our Irish bloodlines a little shaky. Am I descended from a king of Munster, or just from someone who joined the clan? I don't know, and probably never will. The truth is, that particular question probably applies to most people who claim membership of any clan, whether Scottish or Irish. It doesn't affect their membership of the clan, but they may or may not be a descendant of its founder.