The apparent meaning of the author’s words is that if he gives it on the Day of Eid, it counts as payment on time; but if he gives it afterwards, it counts as made up.
The correct view on this matter—and that required by the evidence—is that if he delays it and does not give it until after the Eid prayer, his zakat is not accepted from him. Rather, it is merely one charity among other charities, and he is sinful for doing so.
This is based on the principle indicated by the texts: “Every act of worship that has an appointed time, if a person deliberately performs it outside its time, is not accepted.” For this reason, regarding those who did not pray during the early part of their lives and then Allah graciously guided them, we say that they are not to make up those prayers, because they deliberately allowed the prayer to pass beyond its time—provided that we do not judge them to be disbelievers.
However, if we judge them to be disbelievers, then they do not make them up, because when a disbeliever embraces Islam, he is not commanded to make up what he missed.
Issue: If he delays it due to an excuse—for example, if a person appoints someone to give the zakat on his behalf because he is travelling, then upon returning from his journey discovers that his agent did not do so—then he must make it up, without being sinful, even if the days of Eid have passed. This is by analogy with prayer, due to the Prophet’s statement: “Whoever sleeps through a prayer or forgets it, let him pray it when he remembers it.” (1)
Likewise, if news of Eid comes suddenly and he is unable to deliver it to the poor person until after the Eid prayer, then he is excused; he must make it up and is not sinful.