In regard to your question about the Sunnah of Fajr you mentioned the hadith which was narrated by Umm Mumineen Aisha (Ra) in Sahih Muslim that the Messenger of Allah (ﷺ) said “The two rak’ahs (before) Fajr are better than this world and everything in it.”

“ عَنْ عَائِشَةَ عَنِ النَّبِيِّ صلى الله عليه وسلم قَالَ: ” رَكْعَتَا الْفَجْرِ خَيْرٌ مِنَ الدُّنْيَا وَمَا فِيه “

We can understand from this hadith that the two rak’ahs are referring to the two sunnah of Fajr, consequently that it is bound within the time of Fajr. in other words if a person prays two rak’ahs preceding the time that Fajr comes in regardless of what their intention was, resulting in those two Rak’ahs not being deemed as the Sunnah of Fajr.

Timing of the Sunnah of Fajr

In a Hadith of Aisha (Ra) found in Sahih al-Bukhari and Sahih al-Muslim, Aisha (Ra) reported: The Messenger of Allah (ﷺ) used to perform two Rak’ahs brief prayer between the Adhan (call to prayer) and the Iqama of the Fajr prayer.

عن عائشة رضي الله عنها أن رسول الله صلى الله عليه وسلم، كان يصلي ركعتين خفيفتين بين النداء والإقامة من صلاة الصبح”

In a narration of Sahih al-Muslim mentions that “… When it was Dawn (the time of Fajr) He (ﷺ) would pray two Rak’ahs (sunnah)

“… كان إذا طلع الفجر صلى ركعتين”

In a Hadith of Ibn Umar (RA) in Sunan Nasai Ibn ‘Umar said: “Hafsa (Ra) told me that The Messenger of Allah (ﷺ) used to pray two brief rak’ahs between the call (the Adhan) and the Iqama for Fajr prayer.”

ابْنُ عُمَرَ، قَالَ حَدَّثَتْنِي حَفْصَةُ، أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم: كَانَ يَرْكَعُ رَكْعَتَيْنِ خَفِيفَتَيْنِ بَيْنَ النِّدَاءِ وَالإِقَامَةِ مِنْ صَلاَةِ الْفَجْرِ

In addition to all these Hadith it is mentioned in Sahih al-Bukhari that Aisha (Ra) narrated that The Messenger of Allah (ﷺ) hat offered the Isha prayer (and then got up at the Tahajjud time) and offered eight rak`ahs and then offered two rak`ahs while sitting. He then offered two rak`ahs in between the Adhan and Iqama (of the Fajr prayer) and he never missed them.

عَنْ عَائِشَةَ رضى الله عنها قَالَتْ صَلَّى النَّبِيُّ صلى الله عليه وسلم: الْعِشَاءَ ثُمَّ صَلَّى ثَمَانَ رَكَعَاتٍ وَرَكْعَتَيْنِ جَالِسًا وَرَكْعَتَيْنِ بَيْنَ النِّدَاءَيْنِ، وَلَمْ يَكُنْ يَدَعُهُمَا أَبَدًا

In conclusion from all of these Hadith we can understand that the Sunnah of Fajr is restricted to a set time which is in between the Adhan of Fajr to the completion of the Fardh of Fajr. Although praying two Rak’ahs before the time of Fajr will not constitute as the Sunnah of Fajr. Praying at that time is still considered Tahajjud and is also very fruitful for the believer.

والله اعلم

Allah (SWT) knows best

Zain al-Abidin