The model dependence in the correlations of the neutron-skin thickness in heavy nuclei with various symmetry-energy parameters is analyzed by using several families of systematically varied microscopic mean-field models. Such correlations show a varying degree of model dependence once the results for all the different families are combined. Some mean-field models associated with similar values of the symmetry-energy slope parameter at saturation density L , and pertaining to different families, yield a greater-than-expected spread in the neutron-skin thickness of the 208 Pb nucleus. The effective value of the symmetry-energy slope parameter L eff , determined by using the nucleon density profiles of the finite nucleus and the density derivative S ′ ( ρ ) of the symmetry energy starting from about saturation density up to low densities typical of the surface of nuclei, seems to account for the spread in the neutron-skin thickness for the models with similar L . The differences in the values of L eff are mainly due to the small differences in the nucleon density distributions of heavy nuclei in the surface region and the behavior of the symmetry energy at subsaturation densities.