The above sounds correct for what keeps the Extra Ball relay energized until the next turn starts. As far as what actually gives you the extra ball, you can check this switch:
Extra ball - ball count circuit (resized).png

When the outhole relay fires after a drain, that circuit is what will either advance the ball count and reset the player number, or advance the player number only (depending on whether the last player just drained the ball). But if you have an active extra ball, that switch will open the entire path - keeping the ball and player number unchanged. According to the manual, it's switch C on the Extra Ball relay stack.