Visions are known as the "eye of the gods" and essentially grants the users control over a specific element. They come in different sizes and typically have unique casings depending on the nation the user comes from. There is also a few unique casings for "outlanders" of Teyvat such as the Traveler and the collaboration character, Aloy.
There are multiple theories surrounding the visions but one of the most popular ones is what are the "stipulations" for getting a specific elemental vision. There are multiple versions of these theories that are based on compiling certain characteristics from each character that has that specific element. So let's go through the theories for each vision that a lot of people agree on!
Anemo
Anemo is wind, coming from greek word, Anemos, and is under the control of the Archon Barbatos, god of freedom. One of the most popular stipulations that people tend to agree on for anemo is the loss of a friend or something in their life. Some examples are Kazuha and Xiao, who both lost dear friends to death. Another example is Jean who lost her childhood to pursuing a lofty goal, although she doesn't regret it. Another supposed stipulation that many people have agreed on is that the characters are searching for some type of freedom. Examples include Xiao and Wanderer, who seek freedom from their past, or Lynette who wants the freedom to be with her twin brother, Lyney. While other elements have a lot of different interpretations of the supposed stipulations, this interpretation of Anemo is the one that most of the fandom seems to accept.
Geo
Geo is earth, coming from the prefix of the greek word, Gaia, and is is under the control of the Archon Morax, god of contracts. A popular theoretical requirement that a lot of the fandom tends to agree on is that those with geo visions are those with strong principles. Some examples are Nigguang, who holds to her principles and built herself an empire of wealth from the very bottom, and Chiori, who holds to her principles that the very best service should be given to whoever comes first regardless of status or wealth. A similar theoretical requirement is that geo users typically have high ambitions for themselves. Some examples are Noelle, who seeks to become a Knight of Favonius and so constantly places herself at a high standard to reach that goal, and Albedo, who searches for the truth of the world for himself and seeks to learn more about the world constantly.
Electro
Electro is electricity, coming from the latinized greek word, Elektro, and is under the control of the Archon Beelzebub, god of eternity. One popular theory is that those given electro visions are those that are conisdered outsiders of society. Some prominent examples are Razor, who was raised by the wolves in wolvedom, and Cyno, who is feared by many in Sumeru because of his general position. Meanwhile, there's a different theory that relates more to Beezlebub's ideal of eternity, which is that electro users seek their own form of eternity or stagnation. A few examples are Fischl, who wants to retain her fantasical life and view forever, and Keqing, who has stagnant beliefs of not needing a god to rule over them and ensure their safety and prosperity.
Dendro
Dendro is life, coming from the greek word, Dendron, and is under the control of the Archon Buer, god of wisdom. Dendro is unique game-play wise as it was not released until version 3.0, as such, people have not had as much time to speculate compared to the other elements. However, many dendro characters have been released since the element's release, as such there are quite a few theories. One popular theory is that dendro users seek for some kind of wisdom or are incredibly curious. This was deduced because almost all of the dendro users that the game currently has are all smart in their own way. Examples are that Tighnari, who is a scholar from the Akademyia that is constantly pumping out research papers according to an NPC, and Collei who continues and wants to learn more from Tighnari and about her job as a Forest Watcher. Another theory that people have come up with is that dendro users are obsessive over something. Some examples are Alhaitham, who is constantly searching for knowledge and constantly reading books, and Kaveh who is obsessed with his work as an architect, even to the point of hurting himself.
Hydro
Hydro is water, coming from the greek word, Hydor, and is under the control of the Archon Focalors, god of justice. One theory that people propose for stipulations to being a hydro user is that they often live a masked lie and yet committing full-heartedly to it. This fits in well with the Archon Focalors, as the chapter of Fontaine's archon quest is in fact "the Masked Masqurade", and it is later revealed that the Archon herself is living a lie of sorts. Some other examples are Childe, who often lies to different people, including his family, whether it's for his own gain or to protect them, and yet he refuses to back down from it all, and Mona, who was told in the 1.1 event, Unreconciled Stars, that the sky was fake, and despite being an astrologist with immense faith in the stars, she did not let that information shake her will. Another theory is that the characters are those with high ambitions for people other than themselves. A few examples are Xinqui, who seeks to revive the nearly dead martial arts clan, Guhua, and Barbara's ambition to spreading happiness and love through her songs and prayers.
Pyro
Pyro is fire, coming from the greek word, pur (or the latinized version, pyr), and is under the control of the Archon Haborym, god of war One of the most agreed on "requisites" is that those with pyro visions are incredibely passionate. For example, Diluc is passionate in protecting Mondstadt, especially in secret, and Bennet who was passionate in adventuring depsite his constant bad luck. A similar theory is wanting to walk in the footsteps of their parents or older relatives. Some examples are Hu Tao who follows in the footsteps of her Grandfather, the 75th director of the Wangsheng Funeral Parlor, and Amber who follows in the footsteps of her grandfather, who established the Outriders in the Knights of Favonius.
Cryo
Cryo is ice, coming from the greek word, Kryos, and is under the control of the Archon (according to leaks) Barnabas. A very popular theory is that cryo users are living contradictions. The most prominent examples are Qiqi who is a zombie, both alive and dead, and Kaeya, who is split in his loyalty to Khaenri'ah or to Mondstadt. Meanwhile, another theory is that they are closed off, even from those close to them. Some examples are Rosaria who distances herself from her fellow nuns, and Ayaka who is known as a princess and is separated from the common folk.
Fun fact: almost all of the Archon names are demons from the book of Solomon!
The only exception is the Cryo Archon, who is named after a saint. (unless the leaks are wrong lol)
also I tried to put in greek lettering but I guess this font hates anything other than english haha
check out more theories!