Cell proliferation often experiences a density-dependent intrinsic proliferation rate (IPR) and negative feedback from growth-inhibiting molecules in culture media. The lack of flexible models with explanatory parameters fails to capture such a proliferation mechanism. We propose an extended logistic growth law with the density-dependent IPR and additional negative feedback. The extended parameters of the proposed model can be interpreted as density-dependent cell-cell cooperation and negative feedback on cell proliferation. Moreover, we incorporate further density regulation for flexibility in the model through environmental resistance on cells. The proposed growth law has similarities with the strong Allee model and harvesting phenomenon. We also develop the stochastic analog of the deterministic model by representing possible heterogeneity in growth-inhibiting molecules and environmental perturbation of the culture setup as correlated multiplicative and additive noises. The model provides a conditional maximum sustainable stable cell density (MSSCD) and a new fitness measure for proliferative cells. The proposed model shows superiority to the logistic law after fitting to real cell culture datasets. We illustrate both conditional MSSCD and the new cell fitness for a range of parameters. The cell density distributions reveal the chance of overproliferation, underproliferation, or decay for different parameter sets under the deterministic and stochastic setups.