This article discusses the practical limitations of the flexible cure rate model proposed by Milienos (2022) in ^[1] [On a reparameterization of a flexible family of cure models, Statistics in Medicine 41, 4091–4111]. Although the model is structurally flexible, it has problems with interpretability and identifiability when covariates are included. To address these problems, we propose a novel reparameterization that directly links the cure fraction to meaningful parameters based on cancer cell counting mechanisms. This approach improves interpretability and resolves the identification problems present in the original model. Our new models facilitate straightforward inference within covariate structures while maintaining desirable flexibility. Parameter estimation is conducted using maximum likelihood methods. Through extensive simulation studies, we demonstrate that reparameterized models exhibit superior empirical performance. Their practical applicability is further illustrated through two melanoma datasets, where our models significantly outperform classical approaches in terms of fit and interpretability.