In recent years, phase-field-based models for multiphase flows have gained significant popularity, particularly within the lattice Boltzmann (LB) community. These models typically use two lattice Boltzmann equations (LBEs), one for interface tracking and the other for solving hydrodynamic properties. However, for the purposes of this paper, we focus only on the LB model for hydrodynamics. Our goal is to undertake a comparative investigation into the differences between three classical hydrodynamic LB models proposed by Lee et al. [1], Liang et al. [2] and Fakhari et al. [3]. The interface-tracking equation used in this study is based on the conservative phase-field model. We provide a detailed derivation of the governing equations in each model using the Chapman-Enskog analysis. Additionally, three discretization methods for the interaction forces are introduced, and a modified method for the gradient term is proposed based on the nonequilibrium distribution method. The accuracy of three LB models in combination with four discretization methods is examined in this study. Based on the results, it appears that different combinations of models and methods are appropriate for different types of problems. However, some suggestions for the selection of hydrodynamic models and discrete methods for the gradient term are provided in this paper.