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Aiming at comparing different morphological models of galaxy clusters, we use two new methods to make a cosmological model-independent test of the distance-duality (DD) relation. The luminosity distances come from the Union2 compilation of Supernovae Type la. The angular diameter distances are given by two cluster models (De Filippis et al. and Bonamente et al.). The advantage of our methods is that they can reduce statistical errors. Concerning the morphological hypotheses for cluster models, it is mainly focused on the comparison between the elliptical β-model and spherical β-model. The spherical β-model is divided into two groups in terms of different reduction methods of angular diameter distances, i.e., the conservative spherical β -model and corrected spherical β-model. Our results show that the DD relation is consistent with the elliptical β-model at 1σ confidence level (CL) for both methods, whereas for almost all spherical β-model parameterizations, the DD relation can only be accommodated at 3σ CL, particularly for the conservative spherical β-model. In order to minimize systematic uncertainties, we also apply the test to the overlap sample, i.e., the same set of clusters modeled by both De Filippis et al. and Bonamente et al. It is found that the DD relation is compatible with the elliptically modeled overlap sample at 1σ CL; however, for most of the parameterizations the DD relation cannot be accommodated even at 3σ CL for any of the two spherical β-models. Therefore, it is reasonable that the marked triaxial ellipsoidal model is a better geometrical hypothesis describing the structure of the galaxy cluster compared with the spherica β-model if the DD relation is valid in cosmological observations. [ABSTRACT FROM AUTHOR]