Simplified vehicle models dominate the literature on autonomous driving, with the 3 degrees of freedom (3 DoF) model being the most frequently used due to its high computational efficiency. However, such models have significant limitations, particularly in their inability to account for detailed tire–road interactions. This study proposes an extended model with ten degrees of freedom (10 DoF), developed using the Newton–Euler formalism. Analytical derivation of the mass matrix and the vector of right-hand sides enables significant reduction in computation time by eliminating matrix operations. For comparison, another 10 DoF model based on the homogeneous and joint coordinate transformation method is also considered. The aim is to assess how the choice of modeling formalism affects both computational efficiency and the fidelity of real-world motion representation. All three models (3 DoF and both 10 DoF variants) were tested in simulations of an overtaking maneuver under varying weather conditions. The analysis focused on differences in steering angle trajectories and tracking errors. Additional evaluations included a lane-change maneuver and a 736-meter driving scenario. Results show that extended models provide improved accuracy and better capture of dynamic vehicle behavior. In particular, the Newton–Euler-based 10 DoF model offers significant computational advantages. The maximum observed difference in steering angle between models reached 2 degrees, attributed to the 3 DoF model’s simplified treatment of tire forces and lack of friction coefficient consideration. The proposed models show strong potential for implementation in motion planning for autonomous vehicles.