"As for the two kingdoms sharing a king point, it seems pretty silly. A distinction without a difference,"

Of coure there is a difference. After 1707 Great Britain was a single united kingdom with one monarch and one government. Between 1603 and 1707 they were two seperate kingdoms with two seperate governments. As Head of State the monarch in those days wielded power so sometimes during the 17thC the Stuarts managed to dominate in Scotland from their base in London whilst at other times the Scottish Parliament asserted itself. I have already posted Article One of the said treaty which confirms the winding up of the two seperate kingdoms and the creation of the new united kingdom of Great Britain.