Generally speaking, when using the words ahl al-bayt (people of the house, Arabic: أهل البيت), the wives and daughters are included.
When Allah ﷻ sent the angels over to Ibrāhīm and they gave him and his wife the good tidings of a baby (Prophet Is'hāq), the angels set the mercy of Allah on the people of the house to mean Ibrāhīm and his wife, Sārah:
قَالَتْ يَا وَيْلَتَىٰ أَأَلِدُ وَأَنَا عَجُوزٌ وَهَٰذَا بَعْلِي شَيْخًا إِنَّ هَٰذَا لَشَيْءٌ عَجِيبٌ قَالُوا أَتَعْجَبِينَ مِنْ أَمْرِ اللَّهِ ۖ رَحْمَتُ اللَّهِ وَبَرَكَاتُهُ عَلَيْكُمْ أَهْلَ الْبَيْتِ إِنَّهُ حَمِيدٌ مَّجِيدٌ
She said, "Woe to me! Shall I give birth while I am an old woman and this, my husband, is an old man? Indeed, this is an amazing thing!" They said, "Are you amazed at the decree of Allah? May the mercy of Allah and His blessings be upon you, people of the house. Indeed, He is Praiseworthy and Honorable."
— Surat Hud 11:72-73
In the continuation of the same story, the angels went to the people of Lot and destroyed Sodom and Gomorrah, except for one family and that is of Lot's. The phrase used to describe the family of Lot was āl Lūt (Arabic: آل لوط), which included Lot himself, his wife, and his daughters. His wife was then singled out as the only exception to perish from among āl Lūt:
إِلَّا آلَ لُوطٍ إِنَّا لَمُنَجُّوهُمْ أَجْمَعِينَ إِلَّا امْرَأَتَهُ قَدَّرْنَا إِنَّهَا لَمِنَ الْغَابِرِينَ
Except the family of Lot; indeed, we will save them all Except his wife." Allah decreed that she is of those who remain behind.
— Surat Al-Hijr 15:59-60
As for the verse that talks about ahl al-bayt in the Qur'an, the context of the verses shows that the speech is being address to the wives of the Prophet ﷺ that they are unlike other women, then it passes on six commands to them in order for Allah ﷻ to purify them, then it reminds them to remember the teachings that they come across in their houses:
يَا نِسَاءَ النَّبِيِّ لَسْتُنَّ كَأَحَدٍ مِّنَ النِّسَاءِ إِنِ اتَّقَيْتُنَّ فَلَا تَخْضَعْنَ بِالْقَوْلِ فَيَطْمَعَ الَّذِي فِي قَلْبِهِ مَرَضٌ وَقُلْنَ قَوْلًا مَّعْرُوفًا
O wives of the Prophet, you are not like anyone among women. If you fear Allah, then do not be soft in speech [to men], lest he in whose heart is disease should covet, but speak with appropriate speech.
وَقَرْنَ فِي بُيُوتِكُنَّ وَلَا تَبَرَّجْنَ تَبَرُّجَ الْجَاهِلِيَّةِ الْأُولَىٰ وَأَقِمْنَ الصَّلَاةَ وَآتِينَ الزَّكَاةَ وَأَطِعْنَ اللَّهَ وَرَسُولَهُ إِنَّمَا يُرِيدُ اللَّهُ لِيُذْهِبَ عَنكُمُ الرِّجْسَ أَهْلَ الْبَيْتِ وَيُطَهِّرَكُمْ تَطْهِيرًا
And abide in your houses and do not display yourselves as [was] the display of the former times of ignorance. And establish prayer and give zakah and obey Allah and His Messenger. Allah intends only to remove from you the impurity [of sin], O people of the [Prophet's] household, and to purify you with [extensive] purification.
وَاذْكُرْنَ مَا يُتْلَىٰ فِي بُيُوتِكُنَّ مِنْ آيَاتِ اللَّهِ وَالْحِكْمَةِ إِنَّ اللَّهَ كَانَ لَطِيفًا خَبِيرًا
And remember what is recited in your houses of the verses of Allah and wisdom. Indeed, Allah is ever Subtle and Acquainted [with all things].
— Surat Al-Ahzab 33:32-34
It is obvious that linguistically the entire speech is delivered to the wives, including the verse (إِنَّمَا يُرِيدُ اللَّهُ لِيُذْهِبَ عَنكُمُ الرِّجْسَ أَهْلَ الْبَيْتِ وَيُطَهِّرَكُمْ تَطْهِيرًا). It is important to note that linguistically this does not limit the term ahl al-bayt to only the wives.
The hadith about the cloak you referred to was documented in several books of hadith and through a number of chains of narrations:
قالت عائشة خرج النبي صلى الله عليه وسلم غداة وعليه مرط مرحل من شعر أسود فجاء الحسن بن عليا فأدخله ثم جاء الحسين فدخل معه ثم جاءت فاطمة فأدخلها ثم جاء علي فأدخله ثم قال: إِنَّمَا يُرِيدُ اللَّهُ لِيُذْهِبَ عَنْكُمُ الرِّجْسَ أَهْلَ الْبَيْتِ وَيُطَهِّرَكُمْ تَطْهِيرًا
'A'isha reported that Allah's Apostle (ﷺ) went out one norning wearing a striped cloak of the black camel's hair that there came Hasan ibn 'Ali. He wrapped hitn under it, then came Husain and he wrapped him under it along with the other one (Hasan). Then came Fatima and he took her under it, then came 'Ali and he also took him under it and then said: "Allah only desires to take away any uncleanliness from you, O people of the household, and purify you (thorough purifying)."
— Sahih Muslim, Book 44, Hadith 91
The hadith as is documented in Sahih Muslim neither includes nor excludes the wives of the Prophet ﷺ or any of his other daughters. The hadith has additions through different chains of narration, but none uses an exclusivity form to say the ones under the cloak are exclusively ahl al-bayt. In fact, the hadith of Jubair bin Mut'im extends the concept of the Prophet's family from Bani 'Abdul-Muttalib to Bani Hāshim:
عن جبير بن مطعم، قال مشيت أنا وعثمان بن عفان، فقال: يا رسول الله أعطيت بني المطلب وتركتنا، وإنما نحن وهم منك بمنزلة واحدة. فقال النبي صلى الله عليه وسلم: إِنَّمَا بَنُو هَاشِمٍ وَبَنُو الْمُطَّلِبِ شَىْءٌ وَاحِدٌ
Narrated Jubair bin Mut'im: 'Uthman bin 'Affan went (to the Prophet) and said, "O Allah's Messenger (ﷺ)! You gave property to Bani Al-Muttalib and did not give us, although we and they are of the same degree of relationship to you." The Prophet (ﷺ) said, "Only Bani Hashim and Bani Al Muttalib are one thing (as regards family status).
— Sahih al-Bukhari, Book 61, Hadith 12
In another long hadith narrated through the authority of Yazīd ibn Hayyān shows that his wives are considered among ahl al-bayt (partially quoted), but also specifically included 'Ali, 'Aqīl, Ja'far, 'Abbās, and their offsprings:
ثم قال قام رسول الله صلى الله عليه وسلم يوما فينا خطيبا بماء يدعى خما بين مكة والمدينة فحمد الله وأثنى عليه ووعظ وذكر ثم قال: أَمَّا بَعْدُ أَلاَ أَيُّهَا النَّاسُ فَإِنَّمَا أَنَا بَشَرٌ يُوشِكُ أَنْ يَأْتِيَ رَسُولُ رَبِّي فَأُجِيبَ وَأَنَا تَارِكٌ فِيكُمْ ثَقَلَيْنِ أَوَّلُهُمَا كِتَابُ اللَّهِ فِيهِ الْهُدَى وَالنُّورُ فَخُذُوا بِكِتَابِ اللَّهِ وَاسْتَمْسِكُوا بِهِ. فحث على كتاب الله ورغب فيه ثم قال: وَأَهْلُ بَيْتِي أُذَكِّرُكُمُ اللَّهَ فِي أَهْلِ بَيْتِي أُذَكِّرُكُمُ اللَّهَ فِي أَهْلِ بَيْتِي أُذَكِّرُكُمُ اللَّهَ فِي أَهْلِ بَيْتِي. فقال له حصين: ومن أهل بيته يا زيد؟ أليس نساؤه من أهل بيته؟ قال :نساؤه من أهل بيته ولكن أهل بيته من حرم الصدقة بعده. قال: ومن هم؟ قال: هم آل علي وآل عقيل وآل جعفر وآل عباس. قال: كل هؤلاء حرم الصدقة؟ قال: نعم
He then said: One day Allah's Messenger (ﷺ) stood up to deliver sermon at a watering place known as Khumm situated between Mecca and Medina. He praised Allah, extolled Him and delivered the sermon and. exhorted (us) and said: Now to our purpose. O people, I am a human being. I am about to receive a messenger (the angel of death) from my Lord and I, in response to Allah's call, (would bid good-bye to you), but I am leaving among you two weighty things: the one being the Book of Allah in which there is right guidance and light, so hold fast to the Book of Allah and adhere to it. He exhorted (us) (to hold fast) to the Book of Allah and then said: The second are the members of my household I remind you (of your duties) to the members of my family. He (Husain) said to Zaid: Who are the members of his household? Aren't his wives the members of his family? Thereupon he said: His wives are the members of his family (but here) the members of his family are those for whom acceptance of Zakat is forbidden. And he said: Who are they? Thereupon he said: 'Ali and the offspring of 'Ali, 'Aqil and the offspring of 'Aqil and the offspring of Ja'far and the offspring of 'Abbas. Husain said: These are those for whom the acceptance of Zakat is forbidden. Zaid said: Yes.
— Sahih Muslim, Book 44, Hadith 55
