The Definition to Closed Game, Also Known As: Patch or Clean game, is:
(1) A game with no open frames. This comes from the pursuit of the
fictional USBC award patch for no open frames.
(2) An award given out by the USBC for individual and team
accomplishments. Starting with the 2009-2010 season, the patches have
been replaced by magnets.
According to Wikipedia:
A "Clean Game" is a game with strikes or spares in every frame (not
counting bonus balls).
A single game of bowling where the player has a mark (spare or strike)
in all ten frames.
So according to this definitions the case you describe is in deed a "Closed Game" since you didn't left any open frames. the last 9-pins throw was part of the bonus shoot which isn't included in "Close Game" definition.
Here is a very similar post that present the same question with a little bit different scenario.