The people used to say that Ibn ʿUmar had embraced Islam before ʿUmar. This is not true. What happened is that ʿUmar sent ʿAbdullah to bring his horse from an Ansari man so as to fight on it. At that time the people were giving the Pledge of allegiance to Messenger of Allah ﷺ near the Tree, and ʿUmar was not aware of that. So ʿAbdullah (bin ʿUmar) gave the Pledge of Allegiance (to the Prophet) and went to take the horse and brought it to ʿUmar. While ʿUmar was putting on the armor to get ready for fighting, ʿAbdullah informed him that the people were giving the Pledge of allegiance to Allah's Apostle beneath the Tree. So ʿUmar set out and ʿAbdullah accompanied him till he gave the Pledge of allegiance to Messenger of Allah ﷺ, and it was this event that made people say that Ibn ʿUmar had embraced Islam before ʿUmar.  

أَنَّ ابْنَ عُمَرَ أَسْلَمَ قَبْلَ عُمَرَ وَلَيْسَ كَذَلِكَ وَلَكِنْ عُمَرُ يَوْمَ الْحُدَيْبِيَةِ أَرْسَلَ عَبْدَ اللَّهِ إِلَى فَرَسٍ لَهُ عِنْدَ رَجُلٍ مِنَ الأَنْصَارِ يَأْتِي بِهِ لِيُقَاتِلَ عَلَيْهِ وَرَسُولُ اللَّهِ ﷺ يُبَايِعُ عِنْدَ الشَّجَرَةِ وَعُمَرُ لاَ يَدْرِي بِذَلِكَ فَبَايَعَهُ عَبْدُ اللَّهِ ثُمَّ ذَهَبَ إِلَى الْفَرَسِ فَجَاءَ بِهِ إِلَى عُمَرَ وَعُمَرُ يَسْتَلْئِمُ لِلْقِتَالِ فَأَخْبَرَهُ أَنَّ رَسُولَ اللَّهِ ﷺ يُبَايِعُ تَحْتَ الشَّجَرَةِ قَالَ فَانْطَلَقَ فَذَهَبَ مَعَهُ حَتَّى بَايَعَ رَسُولَ اللَّهِ ﷺ فَهِيَ الَّتِي يَتَحَدَّثُ النَّاسُ أَنَّ ابْنَ عُمَرَ أَسْلَمَ قَبْلَ عُمَرَ