Why does the fast begin with the call to Fajr prayer, contrary to what is stated in the Quran: "And eat and drink until the white thread of dawn becomes distinct to you from the black thread [of night]"? And what is the precise time for the white thread to become distinct from the black thread of dawn, and is it at sunrise as was reported from the predecessors?

What you mentioned, that fasting begins when the white thread becomes distinct from the black thread, is correct. What is meant by it is the whiteness of the day and the blackness of the night, as explained by the Prophet, peace and blessings be upon him, and this is the correct practice that Muslims follow. It is authentically reported in the Two Sahihs that Adi ibn Hatim understood the verse literally by placing two threads, one white and one black, under his pillow. The Prophet, peace and blessings be upon him, corrected him by saying: "It is only the blackness of the night and the whiteness of the day." The eminent scholars Al-Shanqeeti and Al-Qasimi affirm this linguistic interpretation of the verse, and that the Almighty's saying: "of dawn" clarifies what is meant by the white thread and removes any ambiguity. Therefore, your understanding is mistaken, and the practice of Muslims in refraining from things that break the fast from the appearance of dawn is correct. What has been narrated from some of the Salaf regarding the permissibility of eating until sunrise is an ijtihad (independent reasoning) in which they erred. The majority of jurists and the four Imams have gone to the view of abstaining from suhoor (pre-dawn meal) at the appearance of dawn, based on the verse and the prophetic hadiths.