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We discuss results on the dynamics of thermalization for a model with Gaussian interactions between two classical many-body systems trapped in external harmonic potentials. Previous work showed an approximate, power-law scaling of the interaction energy with the number of particles, with particular focus on the dependence of the anomalous exponent on the interaction strength. Here we explore the role of the interaction range in determining anomalous exponents, showing that it is a more relevant parameter to differentiate distinct regimes of responses of the system. More specifically, on varying the interaction range from its largest values while keeping the interaction strength constant, we observe a crossover from an integrable system, approximating the Caldeira-Leggett interaction term in the long range limit, to an intermediate interaction range in which the system manifests anomalous scaling, and finally to a regime of local interactions in which anomalous scaling disappears. A Fourier analysis of the interaction energy shows that nonlinearities give rise to an effective bath with a broad band of frequencies, even when starting with only two distinct trapping frequencies, yielding efficient thermalization in the intermediate regime of interaction range. We provide qualitative arguments, based on an analogous Fourier analysis of the standard map, supporting the view that anomalous scaling and features of the Fourier spectrum may be used as proxies to identify the role of chaotic dynamics. Our work, that encompasses models developed in different contexts and unifies them in a common framework, may be relevant to the general understanding of the role of nonlinearities in a variety of many-body classical systems, ranging from plasmas to trapped atoms and ions.