In fact, literally the only reason it's called "Scots" and not "Inglis", as it originally was, is as the Lowlander Scots gradually developed a sense of national identity separate from the English, they decided that they wanted a national label of their own. But of course, they still didn't want to share a national label or identity with the hated native Celtic-speaking population.
And so "Inglis" became "Scots", while "Scottis" - the native Goidelic language - became "Erse", or Irish.
The whole thing is insidious.