A wide range of meshless methods differ from each other in the process of construction of local approximation, such as moving least squares (MLS), kernel-based, partition of unity (PU), radial basis functions (RBF), and many more. They can be classified into methods based on strong formulation and methods based on weak formulations. One of the oldest meshless methods, the Meshless Finite Difference Method (MFDM), is often typically described as a strong formulation method. However, the method, including its higher-order extensions developed later – the multipoint MFDM and method based on correction terms, has been generalized for various formulations of boundary value problems: strong (local, collocation), weak (global, Galerkin), and mixed global-local ones. Extensions of the MFDM in various weak formulations, including variational ones (Galerkin, Petrov-Galerkin), minimization of the energy functional, and meshless local Petrov-Galerkin (MLPG) have been proposed and examined, with particular focus on MLPG5 variant which yields good accuracy and significantly reduces computational effort due to Heaviside-type test function. This flexibility in formulations is highly valuable for tackling diverse problem types in computational mechanics and PDEs, as each variant aligns with specific application demands – e.g., strong forms for smooth solutions and weak forms for handling discontinuities or complex geometries. Numerical examples illustrate the advantages, disadvantages, and specific aspects of the method's application to different types of boundary value problem formulations.