Sub becomes a Player?

[palmettoref]

I am just want to see if the following question is splitting hairs with the way it is worded??

If a substitute enters illegally, he becomes a player after the ball becomes live? My answer key has this question as T, but the rule book specifically states that a sub becomes a player when the ball becomes live, not after. So which is it --- when the ball becomes live or after it becomes live.
Or would you consider these two occurrences to be at the same time? I am probably reading too much into the question??

Thanks in advance!