Context. In the framework of the hierarchical model the intra-cluster medium properties of galaxy clusters are tightly linked to structure formation, which makes X-ray surveys well suited for cosmological studies. To constrain cosmological parameters accurately by use of galaxy clusters in current and future X-ray surveys, a better understanding of selection effects related to the detection method of clusters is needed. Aims. We aim at a better understanding of the morphology of galaxy clusters to include corrections between the different core types and covariances with X-ray luminosities in selection functions. In particular, we stress the morphological deviations between a newly described surface brightness profile characterization and a commonly used single β-model. Methods. We investigated a novel approach to describe surface brightness profiles, where the excess cool-core emission in the centers of the galaxy clusters is modeled using wavelet decomposition. Morphological parameters and the residuals were compared to classical single β-models, fitted to the overall surface brightness profiles. Results. Using single β-models to describe the ensemble of overall surface brightness profiles leads on average to a non-zero bias (0.032 ± 0.003) in the outer part of the clusters, that is an approximate 3% systematic difference in the surface brightness at large radii. Furthermore, β-models show a general trend toward underestimating the flux in the outskirts for smaller core radii. Fixing the β parameter to 2/3 doubles the bias and increases the residuals from a single β-model up to more than 40%. Modeling the core region in the fitting procedure reduces the impact of these two effects significantly. In addition, we find a positive scaling between shape parameters and temperature, as well as a negative correlation of approximately −0.4 between extent and luminosity. Conclusion. We demonstrate the caveats in modeling galaxy clusters with single β-models and recommend using them with caution, especially when the systematics are not taken into account. Our non-parametric analysis of the self-similar scaled emission measure profiles indicates no systematic core-type differences of median profiles in the galaxy cluster outskirts.