All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.

Dear brother in Islam, may Allah reward you abundantly for your interest in knowing the teachings of your religion and help us fulfill our duties, accept our fasts and righteous deeds, and grant all of us forgiveness by the end of Ramadan.

It is noteworthy that fasting during the month of Ramadan is one of the five pillars of Islam which is of paramount significance. Fasting requires abstaining from food, drink, and sexual intercourse from dawn until sunset. The Prophet (peace and blessings be upon him) said, “Islam is built upon five pillars: testifying that there is no god except Allah and that Muhammad is the Messenger of Allah, performing Prayer, paying the Zakah, making the pilgrimage to the Sacred House (Hajj), and fasting the month of Ramadan.” (Reported by Al-Bukhari and Muslim)

In his response to the question you raised, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:

The time for breaking the fast is the time of sunset. So if the sun sets at 4:56 pm, it is the correct time to break the fast. It is immaterial whether the adhan has been called or not. To wait for the adhan to be called after the sun has set, or to wait for the adhan to finish, is a violation of the Sunnah of the Prophet (peace and blessings be upon him), as he has stated clearly, “My Ummah will always enjoy blessings so long as they hasten to break their fast (as soon as the sun has set).” (Reported by al-Bukhari and Muslim)

The traditions are explicit on the fact that the Prophet (peace and blessings be upon him) was careful to break the fast as soon as the sun had set without any delay whatsoever. Therefore, those who are acting differently are definitely guilty of practicing rigidity which has been condemned by the Prophet (peace and blessings be upon him).

In his well-known book, Fiqh As-Sunnah, Sheikh Sayyed Sabiq states:

It is preferred for the fasting person to hasten in breaking the fast when the sun has set. Sahl ibn Sa`d reported that the Prophet said: “My Ummah will always enjoy blessings so long as they hasten to break their fast (as soon as the sun has set).” (Reported by al-Bukhari and Muslim)

The fast should be broken with an odd number of dates or, if that is not available, with some water. Anas reported: "The Messenger of Allah would break his fast with ripe dates before he would pray. If those were not available, he would eat dried dates. If those were not available, he would drink some water." (Reported by Abu Dawud, al-Hakim, and at-Tirmidhi)

The preceding Hadith also shows that it is preferred to break the fast in the above manner before praying. After the prayer, the person may continue to eat, but if the evening meal is ready, one may begin with that. Anas reported that the Messenger of Allah said: "If the food is already presented, eat before the sunset prayer and do not eat your meals in haste." (Reported by al-Bukhari and Muslim)

We would advise Muslims not to be disturbed or to become divided over differences in such minor issues. Now that you know the correct Sunnah, you can calmly and politely advise those who wait for the adhan to finish, but do not argue over it or let it become a reason to not associate with them.