If a person departs after performing the Tawaf of ‘Umra, will that Tawaf substitute for Tawaf-al-Wada‘?

Narrated 'Aisha:

We set out assuming the Ihram for Hajj in the months of Hajj towards the sacred precincts of Hajj. We dismounted at Sarif and the Prophet ﷺ said to his companions, "Whoever has not got the Hadi with him and likes to make it as 'Umra, he should do it, but he who has got the Hadi with him should not do it." The Prophet and some of his wealthy companions had the Hadi with them, so they did not finish Ihram after performing the 'Umra. The Prophet came to me while I was weeping. He asked me the reason for it. I replied, "I have heard of what you have said to your companions and I cannot do the 'Umra." He asked me, "What is the matter with you?" I replied, "I am not praying." He said, "There is no harm in it as you are one of the daughters of Adam and the same is written for you as for others. So, you should perform Hajj and I hope that Allah will enable you to perform the 'Umra as well." So, I carried on till we departed from Mina and halted at Al-Mahassab. The Prophet called 'Abdur-Rahman and said, "Go out of the sanctuary with your sister and let her assume Ihram for 'Umra, and after both of you have finished the Tawaf I will be waiting for you at this place." We came back at mid-night and the Prophet asked us, "Have you finished?" I replied in the affirmative. He announced the departure and the people set out for the journey and some of them had performed the Tawaf of the Ka'ba before the morning prayer, and after that the Prophet set out for Medina.

حَدَّثَنَا أَبُو نُعَيْمٍ، حَدَّثَنَا أَفْلَحُ بْنُ حُمَيْدٍ، عَنِ الْقَاسِمِ، عَنْ عَائِشَةَ ـ رضى الله عنها ـ قَالَتْ خَرَجْنَا مُهِلِّينَ بِالْحَجِّ فِي أَشْهُرِ الْحَجِّ، وَحُرُمِ الْحَجِّ، فَنَزَلْنَا سَرِفَ، فَقَالَ النَّبِيُّ ﷺ لأَصْحَابِهِ " مَنْ لَمْ يَكُنْ مَعَهُ هَدْىٌ، فَأَحَبَّ أَنْ يَجْعَلَهَا عُمْرَةً، فَلْيَفْعَلْ وَمَنْ كَانَ مَعَهُ هَدْىٌ فَلاَ ". وَكَانَ مَعَ النَّبِيِّ ﷺ وَرِجَالٍ مِنْ أَصْحَابِهِ ذَوِي قُوَّةٍ الْهَدْىُ، فَلَمْ تَكُنْ لَهُمْ عُمْرَةً، فَدَخَلَ عَلَىَّ النَّبِيُّ ﷺ وَأَنَا أَبْكِي فَقَالَ " مَا يُبْكِيكِ ". قُلْتُ سَمِعْتُكَ تَقُولُ لأَصْحَابِكَ مَا قُلْتَ فَمُنِعْتُ الْعُمْرَةَ. قَالَ " وَمَا شَأْنُكِ ". قُلْتُ لاَ أُصَلِّي. قَالَ " فَلاَ يَضُرَّكِ أَنْتِ مِنْ بَنَاتِ آدَمَ، كُتِبَ عَلَيْكِ مَا كُتِبَ عَلَيْهِنَّ، فَكُونِي فِي حَجَّتِكِ عَسَى اللَّهُ أَنْ يَرْزُقَكِهَا ". قَالَتْ فَكُنْتُ حَتَّى نَفَرْنَا مِنْ مِنًى، فَنَزَلْنَا الْمُحَصَّبَ فَدَعَا عَبْدَ الرَّحْمَنِ، فَقَالَ " اخْرُجْ بِأُخْتِكَ الْحَرَمَ، فَلْتُهِلَّ بِعُمْرَةٍ، ثُمَّ افْرُغَا مِنْ طَوَافِكُمَا، أَنْتَظِرْكُمَا هَا هُنَا ". فَأَتَيْنَا فِي جَوْفِ اللَّيْلِ. فَقَالَ " فَرَغْتُمَا ". قُلْتُ نَعَمْ. فَنَادَى بِالرَّحِيلِ فِي أَصْحَابِهِ، فَارْتَحَلَ النَّاسُ، وَمَنْ طَافَ بِالْبَيْتِ، قَبْلَ صَلاَةِ الصُّبْحِ، ثُمَّ خَرَجَ مُوَجِّهًا إِلَى الْمَدِينَةِ.