The language in (A) is a bit abstract, so let's make it more concrete. In the stimulus, for voting to "not be prohibited" means that a necessary condition has not been failed—in other words, that someone is a member in good standing. And for voting to "be authorized" means that a sufficient condition has been met—that someone is guaranteed a vote.

(A) describes the correct flaw—a sufficient-necessary confusion—in abstract language. But by applying (A) to the concrete ideas in the stimulus, we can see that it in fact accurately describes the flaw in Thomas' argument.

Thomas never discusses the character of the club president, meaning he does not commit the ad hominem fallacy that (B) refers to.

Thomas isn't replying to another argument, so it's already unlikely that his flaw will involve misunderstanding someone else. There's only one point where Thomas refers to someone else's claim: he says that Althea admits that properly due-paying members have good standing. But this isn't a mere non-denial, it's an actual admission, so Thomas isn't doing what (C) describes.

There's nothing in the argument to suggest that the club's voting rules are different depending on the issue at question, meaning as far as we know, the issue is irrelevant. Jeffrey either has the right to vote or not, regardless of what he's voting on. Failing to specify something that's not relevant to the argument isn't a flaw of reasoning.

Thomas and Althea don't disagree on the rules of the club, only on whether those rules guarantee Jeffrey a right to vote. Because there's no disagreement regarding the substance of the rules, it doesn't matter whether Althea is an authority. And disregarding an irrelevant consideration isn't a flaw.