If you uttered your niyyah and passed water over your whole body you performed ghusl. Anything else is rather optional or madhhabs based as the qur'an simply says wash yourselves while for prayer washing hands, face and feet and wiping over head is explicitly mentioned:
O you who have believed, when you rise to [perform] prayer, wash your faces and your forearms to the elbows and wipe over your heads and wash your feet to the ankles. And if you are in a state of janabah, then purify yourselves. But if you are ill or on a journey or one of you comes from the place of relieving himself or you have contacted women and do not find water, then seek clean earth and wipe over your faces and hands with it. Allah does not intend to make difficulty for you, but He intends to purify you and complete His favor upon you that you may be grateful. (5:6)
O you who have believed, do not approach prayer while you are intoxicated until you know what you are saying or in a state of janabah, except those passing through [a place of prayer], until you have washed [your whole body]. And if you are ill or on a journey or one of you comes from the place of relieving himself or you have contacted women and find no water, then seek clean earth and wipe over your faces and your hands [with it]. Indeed, Allah is ever Pardoning and Forgiving. (4:43)
and the qur'an is the source for mandatory acts while the sunnah shows what is recommended. So the ruling of a madhhab is primarily based on their interpretation of the verse and the word "washing". And as you should be careful while rinsing mouth and nose during wudu' in ramadan you should be careful doing this while performing ghusl.
On this Ibn Hajar reported in his Bulugh al-Maraam the following hadith:
Allah’s Mesenger (ﷺ) said: “Perform a perfect Wudu run (your fingers) through the fingers of the hands and the toes, and if not fasting, sniff water up well inside the nose”.
[Reported by Al-Arba’a (at-Tirmidhi, abu Dawod, an-Nasa-i and ibn Majah) and Ibn Khuzaima graded it Sahih].
For your information here are the obligations of ghusl according the 4 sunni madhhabs summarized from Sheikh Abd al-Rahman al-Jaziri's book:
- The hanafis hold the opinion that there are three of them 1. rinsing out the mouth 2. rinsing out the nose and 3. washing the entire body with water (this seems to be your source).
In my reference (page 83) it says when describing it for ablution:
As for rinsing out the nose, this involves drawing the water itself inside the nostrils until it reaches the top of the cartilaginous part of the nose. However, the sunnah does not require one to draw the water any further in than this, nor is one to breathe the water in. It is a sunnah-based practice for one to exaggerate the actions of rinsing out one's mouth and nose unless one is fasting, in which case it is undesirable to do so lest one swallow some of the water and unintentionally break one's fast.
- The malikis hold the opinion that there are five of them 1. the intention (niyyah) 2. covering the entire body with water 3. rubbing the entire body during or after pouring water and before the water dries 4. continuity 5. ensuring that water penetrates any and all hair of the body.
- The shafi'is say that there are only two 1. intention 2. covering the outside of the body entirely with water.
- The hanbalis say only covering the entire body with water is an obligation. But they consider for example the inside of mouth and nose as part of the body that needs to be covered.
(You may read more and find more details in Islamic jurisprudence according the four sunni schools volume 1 pages 137-143 for which we have a link here)
For further information refer to:
- Did i need to redo my ghusl?