First of all, the correct title in Scotland is "King" or "Queen of Scots". However outside Scotland they were often referred to as "King" or "Queen of Scotland". So Francis was "Francois, roi d Écosse". Darnley would have been "King of Scots" in Scotland.
After 1603 they preferred to be called "King" or "Queen of Great Britain and Ireland", as if the crowns were united, which of course they weren't till 1707. "Great Britain" was the invention of James VI/I.
I think this is right!