Summary: Many-body one-dimensional systems with \(1/x^2\) interactions are known as Calogero-Moser systems. The so called ordinary system with a common coupling constant for all interacting pairs can be generalized into models with couplings which evolve as additional degrees of freedom. This can be done via unitary reduction of a linear matrix model or, alternatively, by assigning a vectorial degree of freedom to each particle. We briefly review the matrix and vector formulations in the context of Hamiltonian mechanics. Various models of this type were investigated separately in several previous studies. In this paper we focus on relations among them. In particular we prove that the seemingly two-particle variables governing the repulsion of each pair are in fact functions of one-particle variables. Identification of the ordinary Calogero-Moser system as an isolated stationary point in the space of generalized models is a starting point to classifying the generalized models according to the rank of the matrix of couplings or, equivalently, the effective number of independent non-spacial degrees of freedom. Finally, we present two new examples of generalized Calogero-Moser models. The first one combines the matrix and vector degrees of freedom and results in additional \(1/x\) interaction terms. The second one stems from a Hamilton function which mimics the Hamiltonian of a quantum model with spin exchange terms and leads to matrix anticommutator equations of motion.