Scots-Irish does refer to the Ulster Scots in a manner of speaking. Specifically, it refers to the Ulster Scots who migrated to the US.

The term was not so much used at the time of migration, I don't believe. People simply called them Irish. However, I think the term began to be applied to their descendants in the nineteenth century when the US saw a major influx of Irish immigrants (that's "Irish-Irish"). Those already here, descended from the Ulster Scots, described themselves as Scots-Irish to differentiate themselves from the new immigrants. There was a lot of prejudice against the new Irish immigrants at the time, remember.

Signs like these were in store fronts everywhere!

If you are interested in more, I wrote an article on the Scots-Irish immigration for the STM a while back.
http://scottishtartans.org/ulster.html