Throw In or Not
takes the ko. Even if White wins the ko, Black wins by 1 point because of White's captured stone. ( could also be at , with the same result.)
White's best play is , winning the 4 stones. Then Black throws in at and catches up to White with . Now Black can stay even by copying White's plays.
Black to play wins the difference game, while White to play does not. So Black stands at least as well as White. Playing the atari without playing the throw in first is correct. The only possible exceptions require a ko somewhere else on the board.