Rainwater, snow and hail are considered pure because Allah says (in Anfal:11):
وَيُنَزِّلُ عَلَيْكُم مِّنَ ٱلسَّمَآءِ مَآءً لِّيُطَهِّرَكُم
Wayunazzilu AAalaykum mina alssamai maan liyutahhirakum.
And sent down water from the sky to cleanse you.
and again Allah says (in Al-Furqan:48)
وَأَنزَلْنَا مِنَ ٱلسَّمَآءِ مَآءً طَهُورًا
Waanzalna mina alssamai maan tahooran.
We send down pure water from the sky.
Sea Water is deemed pure as per the hadith of Abu Hurairah (Radi Allahu `anhu- رضي الله عنهم) that a man asked the Prophet (Sallallahou Alayhe Wasallam ﷺ ) "O Messenger of Allah, we sail on the ocean and we carry only a little water. If we use it for ablution, we will have to go thirsty. May we use sea water for ablution?" The prophet (Sallallahou Alayhe Wasallam ﷺ ) replied, "Its (the sea) water is pure..." (narrated by Al-Bukhari)
Zamzam Water is pure as per the hadith narrated by Ali (Radi Allahu `anhu) that the Prophet (Sallallahou Alayhe Wasallam ﷺ ) called for a bucket that contained water from the well of Zamzam. He drank from it and then made ablution with it (al-Albani said: It is not from Ahmad’s narration as said in Fiqh Us-Sunnah, rather it is from the additions of his son to him 1/76).
Altered Water (is water that has had its form altered as its been; in the same place for a long period, or the location of it, or the substance mixed with it that can't be removed from it) i.e. water mixed with Algae, tree leaves etc. is categorised under mutlaq water. Because everything that falls under the term water, without any extra Qualifications, is deemed pure. the Quran says (Al-Ma'idah:6):
فَلَمْ تَجِدُوا۟ مَآءً فَتَيَمَّمُوا۟ صَعِيدًا طَيِّبًا فَٱمْسَحُوا۟ بِوُجُوهِكُمْ
Falam tajidoo maan fatayammamoo saAAeedan tayyiban faimsahoo biwujoohikum.
And can find no water, then take some clean sand and wipe your face...
Used Water is purifying no matter who used it. One example of used water that is ritually purifying is recycled water or it is the water that drips from a person who used it to do ghusl or ablution. It is considered pure because it was pure before and has not lost its purity. As narrated by Rubai' bint Mu'wadh who described the ablution of The Prophet (Sallallahou Alayhe Wasallam ﷺ ), "He wiped his head with the remaining water on his hands from his ablution." (narrated by Abu Dawud and Ahmad)
Mixed water can be used as a ritually purifying water as long as its colour, taste and odour are still unchanged (significantly).
If one of the above characteristics has been changed by a pure substance then water becomes "pure" but not ritually purifying meaning water mixed with pure elements can be used for general purposes such as cleaning, cooking...etc. but not for making Wudu or performing Ghusl. Example of pure but not purifying water is rose water.
If one of the above characteristics has been altered by impure substance, aka Najis, then water mixed with impure elements becomes impure.