Eventually these Scotti moved from their homes in Ireland and settled in a cold and rocky place just a few miles across the sea from Ireland and just north of Britannia and founded a kingdom in the west and islands of that place called Dal Riata. Their kingdom was led by kings who, during their coronation, placed a bare foot in a foot-shaped hollow carved out in a flat rock and which can still be seen to this day. Centuries later these former pirates and raiders merged with the Picts, who occupied the lands to their east, to form a new kingdom, which was named after the pirates and raiders - Scotland.
And these Irish settled in Scotland long before the Protestant Ulster-Scots settled in what is now Northern Ireland.
I know it well, since I have some roots in northwestern Ireland and southwestern Scotland. The translation of "Scotti" is new to me, though. It could be that *some* of the Protestant Scots-Irish are descended from them since some of them originated in that area. The Picts still are a mystery. The Scottish people in the NE of Scotland have genes that don't match Gaels in an identical fashion. They seem to be more related to Germanic tribes or Scandinavian groups IIRC. Maybe the Picts were linked to Belgic societies, since some of their (Belgic societies) members evidently settled in England years before the mass waves of Angles, Saxons, Jutes, etc.. It makes sense when you look at the geography.