Praise be to Allah and blessings and peace of Allah be upon the Messenger of Allah and upon his family and companions and those who follow his guidance. To proceed:
Yes, it is as the questioner says. At the time of the Prophet (blessings and peace of Allah be upon him), there was one adhaan, with the iqaamah. When the Prophet (blessings and peace of Allah be upon him) came out to deliver the khutbah and lead the prayer, the mu’adhdhin gave the adhaan, then the Prophet (blessings and peace of Allah be upon him) delivered the two khutbahs, then the iqaamah would be given. This is what is known and this is what is narrated in the books of Sunnah, as the questioner says, and it is something well-known to people of knowledge and faith.
But then the people’s numbers in Madinah increased at the time of the Rightly Guided Caliph ‘Uthmaan ibn ‘Affaan (may Allah be pleased with him), so he decided to add the third adhaan. It is called the first adhaan, and is given to alert the people to the fact that this day is Friday, so that they may prepare themselves and hasten to the prayer before the regular adhaan that is given after the sun has passed the meridian. The Sahaabah followed him in that at his time, such as ‘Ali (may Allah be pleased with him), ‘Abd ar-Rahmaan ibn ‘Awf, one of the ten (who were promised Paradise), az-Zubayr ibn al-‘Awwaam, who was also one of the ten, Talhah ibn ‘Ubaydullah, and other senior Sahaabah. And this is how the Muslims started to do it in most regions, following the action of the Rightly Guided Caliph, ‘Uthmaan ibn ‘Affaan (may Allah be pleased with him); he was followed in that by the Rightly Guided Caliph ‘Ali (may Allah be pleased with him) and the rest of the Sahaabah.
The point is that this happened during the caliphate of ‘Uthmaan and subsequently, and it has remained the practice of most of the Muslims throughout the regions and throughout the centuries until the present day, following the precedent that was set by ‘Uthmaan (may Allah be pleased with him), on the basis of his ijtihaad and sincerity towards the Muslims. And there is nothing wrong with that, because the Messenger (blessings and peace of Allah be upon him) said: “I urge you to adhere to my Sunnah and the way of the Rightly Guided Caliphs; hold fast to it.” He (‘Uthmaan) was one of the Rightly Guided Caliphs, and the interest served by that is quite clear. Hence this was adopted by Ahl as-Sunnah wa’l-Jamaa‘ah, and they did not see anything wrong with it, because it was part of the practice of the Rightly Guided Caliphs ‘Uthmaan and ‘Ali and those of the Sahaabah who were present at that time (may Allah be pleased with them all). End quote.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him)