That's not even close to being accurate except in extremely simplistic terms and even then not very...
I disagree. As history graduate, my simplistic terms though simplistic are fairly accurate. Scotland remained an independent nation until the Act of Union in 1707, we just shared the same monarch from 1603 onwards. Whetever was approved by the English parliament had no effect in Scotland and therefore the execution of Charles I was not approved by any Scots body whatsoever. England therefore not only executed it's own monarch, but a foreign monarch too.
Cromwell had no jurisdiction over Scotland until the invasion after the alliance with the Scottish Covanteers broke down.
EDIT: I also find your dismissive tone slightly offensive.