  We present a comprehensive numerical investigation of non-universal
parameters and corrections related to interface fluctuations of models
belonging to the Kardar-Parisi-Zhang (KPZ) universality class, in d=1+1, for
both flat and curved geometries. We analyzed two classes of models. In the
isotropic models the non-universal parameters are uniform along the surface,
whereas in the anisotropic growth they vary. In the latter case, that produces
curved surfaces, the statistics must be computed independently along fixed
directions. The ansatz h = v t + (\Gamma t)^{1/3} \chi + \eta, where \chi is a
Tracy-Widom (geometry-dependent) distribution and \eta is a time-independent
correction, is probed. Our numerical analysis shows that the non-universal
parameter \Gamma determined through the first cumulant leads to a very good
accordance with the extended KPZ ansatz for all investigated models in contrast
with the estimates of \Gamma obtained from higher order cumulants that indicate
a violation of the generalized ansatz for some of the studied models. We
associate the discrepancies to corrections of unknown nature, which hampers an
accurate estimation of \Gamma at finite times. The discrepancies in \Gamma via
different approaches are relatively small but sufficient to modify the scaling
law t^{-1/3} that characterize the finite-time corrections due to \eta. Among
the investigated models, we have revisited an off-lattice Eden model that
supposedly disobeyed the shift in the mean scaling as t^{-1/3} and showed that
there is a crossover to the expected regime. We have found model-dependent
(non-universal) corrections for cumulants of order n > 1. All investigated
models are consistent with a further term of order t^{-1/3} in the KPZ ansatz.
