In this work, we present a novel approach to bias the driving style of an artificial race driver (ARD) for online time-optimal trajectory planning. Our method leverages a nonlinear model predictive control (MPC) framework that combines time minimization with exit speed maximization at the end of the planning horizon. We introduce a new MPC terminal cost formulation based on the trajectory planned in the previous MPC step, enabling ARD to adapt its driving style from early to late apex maneuvers in real-time. Our approach is computationally efficient, allowing for low replan times and long planning horizons. We validate our method through simulations, comparing the results against offline minimum-lap-time (MLT) optimal control and online minimum-time MPC solutions. The results demonstrate that our new terminal cost enables ARD to bias its driving style, and achieve online lap times close to the MLT solution and faster than the minimum-time MPC solution. Our approach paves the way for a better understanding of the reasons behind human drivers' choice of early or late apex maneuvers.