Multiport network theory has been proved to be a suitable abstraction model for analyzing and optimizing reconfigurable intelligent surfaces (RISs) in an electromagnetically consistent manner, especially for studying the impact of the electromagnetic mutual coupling among radiating elements that are spaced less than half of the wavelength apart and for considering the interrelation between the amplitude and phase of the reflection coefficients. Both representations in terms of Z-parameter (impedance) and S-parameter (scattering) matrices are widely utilized. In this paper, we embrace multiport network theory for analyzing and optimizing the reradiation properties of RIS-aided channels, and provide four new contributions. (i) We offer a thorough comparison between the Z-parameter and S-parameter representations. This comparison allows us to unveil that typical scattering models utilized for RIS-aided channels ignore the structural scattering from an RIS, which is well documented in antenna theory. We show that the structural scattering results in an unwanted specular reflection. (ii) We develop an iterative algorithm for optimizing, in the presence of electromagnetic mutual coupling, the tunable loads of an RIS based on the S-parameters representation. We prove that small perturbations of the step size of the algorithm result in larger variations of the S-parameter matrix compared with the Z-parameter matrix, resulting in a faster convergence rate. (iii) We generalize the proposed algorithm to suppress the specular reflection due to the structural scattering, while maximizing the received power towards the direction of interest, and analyze the effectiveness and tradeoffs of the proposed approach. (iv) We validate the theoretical findings and algorithms with numerical simulations and a commercial full-wave electromagnetic simulator based on the method of moments.