The Koopman framework proposes a linear representation of finite dimensional nonlinear systems through a generally infinite dimensional globally linear representation. Originally, the Koopman formalism has been described for autonomous systems and the extension for actuated continuous-time systems with a linear input or a control affine form has only recently been addressed. However, such a derivation for discrete-time systems has not yet been developed. Thus, a particular Koopman form is generally assumed, predominantly a linear time invariant (LTI) model, as it facilitates the use of control techniques such as linear quadratic regulation and model predictive control. However, we show that this assumption is insufficient to capture the dynamics of the underlying nonlinear system. In the present paper, we systematically investigate and analytically derive lifted forms under inputs for a general class of nonlinear systems in both continuous and discrete time, using the fundamental theorem of calculus. We prove that the resulting lifted representations give linear state-space Koopman models where the input matrix becomes state (and input, for the discrete-time case)-dependent, hence it can be seen as a specially structured linear parameter-varying (LPV) description of the underlying system. We also provide error bounds on how much the parameter variation contributes and how well the system behaviour can be approximated by an LTI Koopman representation. The introduced theoretical insight greatly helps for performing proper model structure selection in system identification with Koopman models as well as making a proper choice for LTI or LPV techniques for the control of nonlinear systems through the Koopman approach.