All perfect praise be to Allah, The Lord of the Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
It is an act of the Sunnah for whoever hears the Azan to repeat it and recite the supplication due afterwards, which is: " O Allaah, Lord of this perfect call and of the regular prayer which is going to be established. Kindly give Muhammad the right of intercession and superiority and send him (on the Day of Judgment) to the best and highest place in Paradise which You promised him), " [Al-Bukhari]. This is of course if the person has time to do so. Afterwards, he should perform prayer in congregation or individually, if he couldn't.
Moreover, when the Muezzin ( A man who calls Muslims to prayers from a Mosque) calls " Allah Akbar" then the time of prayers is announced, and it is permitted to be followed by the Opening Takbir immediately. In addition, it is not a condition that the praying person(Who is at home) should wait until the Mosque's Iqamah is called, but he may start performing prayer immediately when the time of prayer is announced.
However, what you (Asker) read on the website reflects people`s impression, in some countries, that Dawn Azan is called before the daybreak. Nonetheless, this doesn't apply on all Muslims due to their countries' geographical locations and different timings.
Yet, the prayer timings in our country (Jordan) are set by a specialized Sharia Board which has reviewed timings and made use of astronomical calculations then documented the results in a book printed by the Ministry of Awqaf and Islamic Affairs in 1982 AD.
Therefore, there is no harm for a Muslim to abide by them (Prayer timings) and whoever has some notes to add in this regard should inform the Ministry about them. However, their opinions/notes are binding to none but themselves in case they are actually specialized in timings. And Allah knows best.