I think that when considering the Scots-Irish / Ulster Irish we should be thinking of region more than nationality, and we should be considering culture rather than ethnicity. Also remember that while many came to North America by way of northern Ireland,others migrated straight from Scotland. Further, consider the effects of generations of life in Ulster, and after a while it becomes silly to think of these people as Scottish, English, or Irish exclusively.
Historian David Hackett Fischer refers to these migrations as "northern British" in character, and that's probably as accurate as any other label.