@AuthorMonika No; we have been British here long before we were Scottish. British comes from 'Pritani' which is what the people here called themselves and the whole island before the Romans invaded.
When the Romans were here they Latinised this to Britannia, but also, for example, the Vindolanda tablets found at Hadrian's wall record the locals north of the wall as 'Britunculi' ' the little britons'
Looking at the post Roman period the 'scots' were Irish pirates who started to invade and colonize from the west. There has been a tribal split known between the 'British' - which covered the central belt where most of 'Scots' still live now, and the 'picts' to the North.
Now the latest research on the Picts suggests from their language that they and the 'British' had an originally common source and both from a same root proto-british language.
Then if you move forward to the Declaration of Arbroath - it literally says that the Scots invaded from Ireland and 'expulsis primo Britonibus et Pictis omnio deletis' - 'first routed the British then destroyed every single Pict'