In this article, I present different definitions of elementary processes which lead to definitions of Itô integrals against cylindrical Wiener processes. I prove the equivalence between those definitions by constructing this Itô integral using only Itô isometries and extensions by density. Then, from the perspective of white noise theory and Kondratiev spaces, I compare the definitions of cylindrical Wiener processes, and present a link between Itô integral and cylindrical white noise.