500 years ago, there was a new hydro archon that came after Egeria's passing, this new hydro archon's name was Focalors and would look like this:

In a old art styled cutscene

In-game model

The new hydro Archon Focalors would come up with a plan to deceive the heavenly principals, but to do so, she split herself into a divine being and a human being..this human's name was Furina. Focalors told Furina that she would have to play as a god for a very long time to save her people from the prophecy, Furina was definitely nervous at first but would soon speak all bratty and high and mighty to sell the deal with her people that she is truly the Hydro Archon.

But, her act would soon end when she was sent to trial and received the death sentence, but it wasn't Furina that was dying..it was Focalors, and since Focalors has died, Furina was finally able to rest..being able to act as a human, just like Focalors wished she could.

Why she has a vision: Since Furina is a human side of Focalors and not a real god, she was able to officially gain a vision and not use a fake one like Venti and Zhongli.

From her character story quest: 

more about her;
The real Furina:

The act of the Hydro Archon (has same long hair as Focalors) :

the reason why Furina is terrified of Arlecchino: Despite being cursed, she had no real powers because she was simply human and not a real god

reason of the shadow Furina in the first Fontaine quest: this was the real her aka, her without the act, but she can't show the real her in front of her own people, so this was simply her in her head.