In recent years, numerous lattice Boltzmann Methods (LBM) for the simulation of multi-phase flows have been proposed, especially tackling the simulation of weakly compressible flows in the presence of a dynamic interface, where surface tension plays a key role. On the other hand, a large spectrum of reduced-order approaches is available in computational fluid dynamics (CFD) to model multi-phase flows, both in the compressible and incompressible regimes. They range from multi-fluid models, eventually including sharpening techniques, to second-gradient theory applications. These reduced order models are then discretized by means of classical numerical methods (Finite Volumes, Finite Elements, etc.) The aim of the present study is to compare lattice Boltzmann models with such reduced order models and numerical strategies developed by the CFD community in order to shed some light on the level of modeling attained by the LBM strategies. We will also rely on some theoretical background based on recent advances in the field of LBM clarifying the notions of consistency and stability in the relationship with multi-step finite difference formulations.