Vision Headcanons
So based off of the characters I've played/read about/etc. I have a few theories on how certain characters are given certain visions. This is also based off some other theories people have had around the site.
PYRO VISIONS
The Vision of Passion
Those given this vision have showed great passion for the things they strive to do. They have goals or desires they want so badly that it is like a flame in their heart that cannot be extinguished. It is something that they will never give up on.
They are given a Pyro Vision due to this.
GEO VISIONS
The Vision of Perseverance
One could also say it is the Vision of Hard Work. Being given a Geo Vision is a symbol of one's ability to go above and beyond in their field. Unlike the Vision of Passion, this one has a grander goal with much more care and effort put into it. Those with this have already reached great heights with their efforts (including Noelle, as she is well known around Mondstadt to be able to handle anything).
Those who have had their hard work paid off may receive a Geo Vision in return.
ANEMO VISIONS
The Vision of Rest
This Vision is meant to come as a symbol to its receiver to rest. It is often received by those who are unable to free themselves from the burden of their duty whether self-inflicted or not. It is to say "you've done well, please rest for a time." The Anemo Archon is known to be also one of freedom so that his vision may also represent free will.
Being given it may be a reminder of the free will they have to take a break.
CRYO VISIONS
The Vision of Love
Those with this Vision have faced an incredible hardship in their life, one that could have been avoided if only love had been there to warm them. The Cryo Archon is unable to give that love as she is the embodiment of ice and cold. A vision is the most one can give to acknowledge the receiver's lack of love.
A Cryo Vision may well be given to those who have not received the warmth of another in a long time.
HYDRO VISIONS
The Vision of Childhood
Like Cryo Visions, this is also meant to be show the lack of its meaning. Receiving a Hydro Vision means that you have faced something that had stripped you of a proper childhood. Whether it was the burden of responsibility, an unfortunate accident that forced you to be more mature, or otherwise missed out on the wonders of youth, a Hydro Vision is bound to be in your hands.
ELECTRO VISIONS
The Vision of Strength
This has been one of the harder ones to figure out. I like to believe that it will be better explained once we are able to travel to Inazuma and learn of the possible reasons for the Electro Archon to want to take back her Visions.
Receiving an Electro Vision is a symbol of strength to any who may catch a glance of them. It is not simply in physical strength that they are strong but emotional and psychological strength as well. Being able to push forth with their lives. They show great strength in both their personal and professional lives.
There is overlap between all of these which I believe is purposeful. Certain Archons may be able to grant visions to the same character but it is a matter of who believes they are best to represent themselves. There are so little Geo users which may be indicative of Zhongli's reserved personality. There are also not many Anemo Users as of right now, showing how perhaps Venti does not pay as much attention to the people of Teyvat (much more preoccupied with escaping from his own past).
As we learn more about the world of Teyvat and the Archons in it, we'll better understand these Visions and what they represent.