What is the ruling on someone who ate, thinking that it was still the time of the Adhan, then discovered that it was between the two Adhans? And does the ruling differ in Ramadan from other times?

The determining factor for abstaining is the rise of the true dawn (Fajr As-Sadiq), based on the Almighty’s saying: "And eat and drink until the white thread of dawn becomes distinct to you from the black thread [of night]." The second Adhan (definition) is the one that occurs at the rise of dawn, based on the Hadith (definition) of Ibn Umar, in which the Prophet, peace and blessings be upon him, said: "Indeed, Bilal calls the Adhan at night, so eat and drink until Ibn Umm Maktum calls." So, if the first Adhan in the mosque is at the rise of dawn, then one must abstain, and the fast is valid. However, if it is before the rise of dawn, then it is permissible to eat and drink until the rise of dawn is ascertained.