Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.

I would like to shed light on the authenticity of the variant version of the hadith you have just quoted and then explain what does it mean. This hadith has been reported in two slightly different wordings.

The first version gives mention to the family of the Prophet as one of the two primary sources a Muslim must adhere to. It has been reported in different versions and narrations through different companions of the Prophet one of those narrations has been related by Zayd b. Arqam who reported the Prophet as saying, “Indeed, I am leaving among you, that which if you hold fast to them, you shall not be misguided after me. One of them is greater than the other: The Book of Allah is a rope extended from the sky to the earth, and my family – the people of my house – and they shall not split until they meet at the Hawd, so look at how you deal with them after me.” (At-Tirmidhi)

According to the narration of Imam Muslim, it reads,  “One day Messenger of Allah (peace be upon him) stood up to deliver a Khutbah at a watering place known as Khumm between Makkah and Al-Madinah. He praised Allah, extolled Him and delivered the Khutbah and exhorted (us) and said, ‘Amma Ba’du (now then)! O people, certainly I am a human being. I am about to receive a messenger (the angel of death) from my Rubb and I, in response to Allah’s Call, but I am leaving among you two weighty things: 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.”‘ (Muslim)

A third version has been reported by Jabir b. Abdullah who saw the Messenger of Allah during his Hajj, on the Day of `Arafah when he was upon his camel Qaswa, giving a Khutbah and saying: ‘O people! Indeed, I have left among you, that which if you hold fast to it, you shall not go astray: The Book of Allah and my family, the people of my house.'” (At-Tirmidh)

Other narrations give mention to the word ‘Sunnah’ referring to the Prophet’s normative way of life instead of his family members. This hadith has been related through different narrations from various Companions of the Prophet.

It reads as follows: “I have left among you two matters by holding fast to which, you shall never be misguided: the Book of Allah and the Sunna of His Prophet.”

It has been narrated from Anas as related by Abu al-Shaykh in Tabaqat al-Muhaddithin fi Asbahan (4:67 p. 549); also from `Amr ibn `Awf  as related by Ibn `Abd al-Barr in al-Tamhid (24:331); from Ibn `Abbas as related by Ibn Nasr al-Marwazi (202-294) in al-Sunna (p. 25-26, no.: 68) and al-Hakim in his Mustadrak (1:93) who declared that all its narrators are “agreed upon” meaning in the two books of Sahih. It has been further related by Al-Bayhaqi (10:114 no.: 20108) and many others.
Those two hadith do not conflict with one another. Proper understanding of both hadith may be summarized in the following points:

1. The majority of scholars are of the view that the household which Muhammad is referring to here include not only `Ali, Fatima and the descendants of Muhammad, it includes his wives as well. When we look for the word “ahlulbayt” (meaning household) in the Qur’an, then we find that it has been mentioned in Chapter 33 (Ahzab), verse 33. ” Allah intends only to remove from you the impurity, O people of the [Prophet’s] household, and to purify you with [extensive] purification”. Ahl Al-Bayt also include the descendants of Hashim either from the side of `Ali or Al-Abbas, the descendants of al-Harith and the progeny of Abu Talib. Those are the ones who are prohibited to receive zakah.
2. The first report commands the Prophet’s companions to adhere to the book of Allah as a source of guidance and he recommended them to revere, love and respect the members of his family. This meaning has been confirmed by many exegetes of the Quran such as Qurtubi and ibn Katheer.
3. The hadith may be taken as an evidence for the authority and applicability of consensus sought from the progeny of the Prophet including the categories of Ahl al-Bayt.
4. The hadith does not mean that we exclusively rely on scholars who descended from the Prophet and his family while abandoning the legacy of other scholars.

Allah Almighty knows best.