Now that the black mistake in the original sequence has been explained by Laval, maybe it's interesting to give an analyse of the original position. It's not very easy to tell who will be able to play first at a.
$$ Initial position
$$ -.......
$$ -.......
$$ -...OOOO
$$ -.....XO
$$ -.OOOOOX
$$ -XOXXO.X
$$ -aX..XX.
$$ -.......
$$ --------
If the temperature of the board is sufficiently low (<6 pts) then a is a double sente and should be played as soon as possible by both players. (Net gain will be 1 point if black is ko master, but up to 3 points if he is not)
However let's have a look at the continuations if the other player does not reply:
$$B Black continuation, 7 points
$$ -.......
$$ -46.....
$$ -32.OOOO
$$ -578..XO
$$ -1OOOOOX
$$ -XOXXO.X
$$ -XX..XX.
$$ -.......
$$ --------
and:
$$W White continuation, about 6 points (12 points gote, assuming black is alive)
$$ -.......
$$ -.......
$$ -...OOOO
$$ -.....XO
$$ -.OOOOOX
$$ -XOXXO.X
$$ -OX12XX.
$$ -.3.....
$$ --------
Black continuation is a lot bigger than white's one. (Sorry in my previous count, I miscounted.)
Black continuation is only slightly bigger than white's one. This means that both players might have the opportunity to play at a first. Black might play it while the temperature of the board is between 7 and 6 (that is, before it strictly becomes double sente), but white will only play it below 6. So, black having the opportunity to play a first is a little bit more probable, and it might be taken into account when estimating the score.