The paper considers the model predictive current control (MPCC) of an induction motor (IM) drive and evaluates five IM models of different complexities—from conventional to magnetic saturation, iron losses, and stray-load losses—for the MPCC design. The validity of each considered IM model and the corresponding MPCC algorithm is evaluated by comparison of the following performance metrics: the total harmonic distortion of the stator current, the average switching frequency, the rotor flux magnitude error, the rotor flux angle error, and the product of the first two metrics. The metrics’ values are determined in wide ranges of the rotor speed (0.1–1 p.u.) and load torque (0–1 p.u.) through simulations performed in the MATLAB Simulink environment. The obtained results allow us to identify the IM model that offers the best tradeoff between the practicability and accuracy. Furthermore, a control effort penalization (CEP) is suggested to reduce the average switching frequency and, hence, the power converter losses. This involves constraining the simultaneous switching to a maximum of two branches of the three-phase power converter, as well as inclusion of the weighted switching penalization term in the cost function. Finally, the performance—both steady-state and dynamic—of the proposed MPCC system with CEP is compared with that of the analogous field-oriented controlled (FOC) IM drive. The inverter switching frequency is reduced more than twice by including the frequency-dependent iron-loss resistance in the MPCC. It is additionally reduced by implementing the proposed CEP strategy without sacrificing many other performance metrics, thus achieving a performance comparable to the FOC IM drive.