Rinsing the Mouth and Nose in the Ritual Bath

The disagreement

There is no disagreement regarding the obligation of performing a ritual bath (ghusl) due to major ritual impurity (janabah) for everyone required to pray.
They disagreed on whether rinsing the mouth and snuffing water into the nose are conditions for the validity of the ritual bath. This disagreement consists of two opinions.

The opinions

Rinsing the mouth and snuffing water into the nose are not mandatory in the ritual bath:
Held by: Mālik and al-Shāfiʿī.
Rinsing the mouth and snuffing water into the nose are mandatory in the ritual bath:
Held by: Abū Ḥanīfah and Aḥmad.

Reason for the disagreement

The contradiction between the ḥadīth of Umm Salamah رضي الله عنها and the ḥadīths narrated describing the manner of the Prophet's ﷺ ablution for his purification.

Evidence

For the first opinion:

The ḥadīth of Umm Salamah رضي الله عنها: "It is sufficient for you to pour three handfuls of water over your head, then pour water over yourself, and you will be purified." [Agreed upon].
The general implication of the saying of Allah the Exalted: ﴿وَإِن كُنتُمْ جُنُبًۭا فَاطَّهَّرُوا۟﴾ "And if you are in a state of janabah, purify yourselves" [Al-Mā'idah: 6], which does not explicitly mention rinsing the mouth or snuffing water into the nose.

For the second opinion:

The ḥadīth of ʿĀʾishah رضي الله عنها: "When the Messenger of Allah ﷺ performed ritual bathing from janabah, he would start by washing his hands, then pour water with his right hand over his left hand to wash his private parts, then perform the full ablution (wudu) like that for prayer."
This ḥadīth clarifies the general command found in the ḥadīth of Umm Salamah and serves as an explanation for the text of the verse: ﴿وَإِن كُنتُمْ جُنُبًۭا فَاطَّهَّرُوا۟﴾. Allah commanded the purification of the entire body, and the inside of the mouth and nose cannot be exempted from this purification unless it is impossible to wash them.

References

Bidāyat al-Mujtahid wa Nihāyat al-Muqtaṣid (1/93), al-Hidāyah (1/60), Badā’iʿ al-Ṣanā’iʿ (1/267), al-Tafrīʿ (1/191), al-Maʿūnah (1/17), al-Bayān (1/253), Rawḍat al-Ṭālibīn (1/199), al-Sharḥ al-Kabīr (1/325), al-Mughnī (1/291), Sharḥ Ibn Zāḥim (1/290).