You're correct - all of us are British, but English people have the habit of referring to the UK as England. Since the time of Thatcher, much resentment was caused in Scotland by her government's anti Scottish policies, leaving us all now with a drift away from the the UK. "Britain", as you say, means also the "UK." And with any smaller country adjacent to a larger one, you tend to get people being assertive about their culture etc. Scotland has 5 million to England's 60.

All have the same passport, a British or UK passport.

Braveheart, I'm sorry to say, is Tartan Hollywood, portraying the "evil" English, and the "brave" Highlanders. It follows the cliches of a Western movie. None of this is accurate.

Incidentally, Scotland holds two main cultures - the Highland area - initially Gaelic speaking, and Lowland Scotland, which always spoke what we now call English (Northern English) since varieties of that language formed in these islands. The Scots were historically hostile to the Highlanders, and were part of the army which routed them at Culloden. Scots kings passed a number of laws against Highlanders.

After the Highlanders were defeated, many emigrated to, for example, the US, or Australia, where they inflicted the same horrors on Native Americans or Aborigines, that were inflicted on them.