Different notions of identifiability are available for discrete-time nonlinear systems. The most popular in the current literature is structural identifiability, besides algebraic identifiability and identifiability with known initial conditions. These three notions are fully characterized in this paper within a unique linear algebraic framework. The relationships between the notions are shown as well. Academic examples as well as the model of a power amplifier, borrowed from the area of digital telecommunications, are used to illustrate the different notions and computations involved by our results.