Does fasting become invalidated if I thought that Fajr had not yet begun and I drank, then I heard an adhan from a distant area, and after that the muezzin of my neighborhood called the adhan? And what if the adhan was called and I drank after the adhan?

The validity of the fast is determined by abstaining from things that break the fast from the rise of true dawn, or upon hearing the adhan from a reliable muezzin who depends on a trusted calendar. If the muezzin relies on a trusted calendar, then the fast is valid. However, drinking after the adhan, when it is known that it is after the rise of true dawn, invalidates the fast and necessitates making up that day.