In fact, both names are rights. Regularization is the term used when the model admits an additional term during learning (sum theta for L1 and sum theta squared for L2). This term penalizes the model during training to reduce overfitting. MSE or MAE is the name given for the cost function which measures the distance between the real data and that predicted by the model. The two terms correspond to the same thing but are not calculated in the same way in the same place. Regularization is computed inside the NN and the cost function at the end (permits to give information to the previous layers with backpropagation).