Are the "People of the Prophet" in the Hadith of Thaqalayn intended to be Fatimah, Ali, Hasan, and Husayn?

Scholars have differed in defining the Household (Ahl al-Bayt) of the Prophet, peace be upon him. The preponderant view is that the Prophet's wives are included in his Household, based on the Almighty's saying: "Allah only intends to remove from you the impurity, O people of the Household, and to purify you with a thorough purification" (Quran 33:33). Also, Banu al-Muttalib are included in his Household, based on the Prophet's saying, peace be upon him: "Indeed, Banu al-Muttalib and Banu Hashim are one and the same." Included in the Household are Banu Hashim, who are the families of Ali, Abbas, Ja'far, Aqil, and Al-Harith ibn Abd al-Muttalib, because they are among those for whom sadaqah (charity) is forbidden. As for the mawali (freed slaves), they are included in the Household based on the Prophet's saying, peace be upon him: "Indeed, we, the family of Muhammad, are not permitted to accept sadaqah, and the mawla of a people is one of them." Thus, the Household of the Prophet, peace be upon him, consists of his wives, his descendants, Banu Hashim, Banu al-Muttalib, and their mawali.