C.S. Lewis used the nickname Jack for himself after childhood, rather than his given name Clive.
The nickname developed after the death of the Lewis family dog, Jacksie, when Lewis was a child. According to accounts of his early life, he announced that he would no longer answer to the name Clive and would instead be called Jacksie. The form was later shortened to Jack.
Jack became the name used by friends, family, and many colleagues throughout his adult life. It appears frequently in biographies and in the correspondence of people who knew him personally. His brother Warren Lewis, usually called Warnie, was among the people closest to him.
“Tollers” is a different nickname: it was used for J.R.R. Tolkien, Lewis’s friend and fellow Oxford academic. “Jacksie” belongs to the story of how Lewis arrived at Jack, but Jack is the established adult nickname. The distinction matters because C.S. Lewis’s public initials and his private nickname refer to the same person.