Is the large size of the crescent moon on the first night considered one of the signs of the "inflated crescents" mentioned in the Hadith, and what is the criterion for that? And does seeing it large on the first night mean that the fast of the preceding day was incorrect?

The hadith (definition) "inflation of the crescents," which is mentioned as one of the signs of the Hour, does not have an authentic chain of narration that can be relied upon.

The various chains of narration through which the hadith has been transmitted are weak. In some of them, there are unknown narrators; in others, there is weakness in precision or abandonment of the hadith; and some are mursal (lacking a Companion in the chain).

In addition to the weakness of its chains of narration, making "inflation of the crescents" a sign of the Hour is questionable, because the largeness of the crescent and its appearance bigger than its first night is an ancient and observable phenomenon. People may differ in their estimation of it, and the Prophet, peace be upon him, and Umar ibn al-Khattab, may Allah be pleased with him, indicated that the size of the crescent, whether large or small, has no legal consideration in determining the beginning of the month or its rulings.

The jurists have not relied on the largeness or smallness of the crescent in any legal rulings; the consideration is based on sighting or completing the number of days.