Is drinking water after hearing an adhan based on assumption—and it later became clear that it was a moment before the entry of the time—considered negligence that necessitates making up the fast, especially given that the correct adhan was broadcast moments later?

Your fast is valid as long as you believe you did not drink after the time for Fajr entered. At the very least, you would have been doubtful about its entry. And whoever eats or drinks while doubtful about the entry of the time for Fajr, their fast is valid, even if it is preferable for them to abstain. Some scholars have even stated the validity of the fast for one who ate or drank a minute or so after the adhan, if the muezzin made the call to prayer based solely on the calendar.