But there is a difference in the definitions. In the former it's said that "the image of the subset is the same as the image of the whole space," but in the other it's said symbolically that the image of the subset is the entire codomain space.
It's more credible that the terminology of the paper is the correct, but in some papers is used nonstandard terminology and I'm a little bit confused. What is the meaning of the term "inductively P map" is the sense of the paper standard or is the other page correct? and in any case what is the terminology of the other case?