You are correct. "The king is dead, long live the king" is the original version - it comes from the legal concept that the King never dies - just Henry VIII or Mary II or whichever sovereign person is holding the office at the time.
So, one king has died, but the next ascends to the throne immediately (this is the case in most, though not all, European monarchies). The position of King, however, is never empty.
It's also a simulateous way to mourn the dead Monarch and celebrate the new one.
King or Queen can be used in either place, depending on the sex of the Monarch.
When Elizabeth II's grandfather died, it would have been, "The king is dead, long live the king," when her father died, "The king is dead, long live the queen," and when she passes it will be, "The queen is dead, long live the king."
You weren't misinformed
So, one king has died, but the next ascends to the throne immediately (this is the case in most, though not all, European monarchies). The position of King, however, is never empty.
It's also a simulateous way to mourn the dead Monarch and celebrate the new one.
King or Queen can be used in either place, depending on the sex of the Monarch.
When Elizabeth II's grandfather died, it would have been, "The king is dead, long live the king," when her father died, "The king is dead, long live the queen," and when she passes it will be, "The queen is dead, long live the king."