Still think that's describing it a little more mildly than was actually the case.
The 2 had been fighting for centuries; and the only thing that made the first union even possible was a Scot king marrying into the English royal family and his descendant ultimately ascending to the English throne. Even then they were still a separate state.
The only group of Scots that had anything to gain from a new union, following James' overthrow, were the wealthy nobles and landowners; and since they essentially made up the parliament, they made a decision that was self-serving despite overwhelming opposition from the population.
Additionally, England was threatening yet another invasion unless Scotland signed the treaty. So, "of their own free will" or "of their own accord" is a little disingenuous since there was a caveat.
Making a decision of your own free will and making a decision of your own free will because there's a gun to your head aren't quite the same thing.