Censored World was only introduced on the 2nd light novel, Extra Stories of Psychics 2. It was the result of when Kusuo didn't enroll on PK Academy and then wasn't able to mingle and talk to the other students/characters and then turned the world into a censored world in just the 10th chapter. Instead, the other work of the author, the Mikami Case Files was serialized.
Modified 2014 World
Modified 2014 was first introduced in the Volume 12 of the series (Season 2, Episode 2 in the anime.) The world was triggered when Kusuo influenced the meeting of his parents encounter which changed the flow of the events. Kusuke created a time machine in order to bring their Kusuo back in their world but it caused for the World War 3 to break out and the world met its destruction. The cause of Kusuo's death was unclear but in the Modified 2014 World, he has been dead for more than 3 years.
Unable To Save The Dog World
This world was first introduced in Volume 7. Kusuo wakes up and sees numbers with his name above his head (along with everyone else), and discovers that now he is very popular and that everyone knows him. He (at that point transformed into Kusuko, making him a "she") then met the Saiki Kusuo from that world and he explains how the world was created. In Chapter 1 or Episode 1, in this world "Kusuo" wasn't able to save the dog from the incoming truck. Devastated by the accident and his hesitant attitude, "Kusuo" decided to change for the better and used his psychic powers as people came to ask for his help.
This world was created based on the Official Poll results.
Worlds Created by Altering the Past with Touma
In order to conceal himself as being a psychic from Touma, Kusuo creates multiple parallel worlds after various efforts using his time leap ability. Even a slight modification had a major effect due to the butterfly effect. Kusuo reveals in one of them that he is a psychic to his classmates.
A Different World After The Volcano
After ending the time loop, the others witness Kusuo using his abilities. They were in shock and asked Kusuo several questions. Kusuo does not answer most of them and states he will not tell them anything yet. They all decide to hear him out once he is ready. After the others got distracted, Kusuo time travels prior to the incident and successfully stops the eruption without getting exposed.