Cutting state monitoring in the milling process is crucial for improving manufacturing efficiency and tool life. Cutting sound detection using machine learning (ML) models, inspired by experienced machinists, can be employed as a cost-effective and non-intrusive monitoring method in a complex manufacturing environment. However, labeling industry data for training is costly and time-consuming. Moreover, industry data is often scarce. In this study, we propose a novel adversarial domain adaptation (DA) approach to leverage abundant lab data to learn from scarce industry data, both labeled, for training a cutting-sound detection model. Rather than adapting the features from separate domains directly, we project them first into two separate latent spaces that jointly work as the feature space for learning domain-independent representations. We also analyze two different mechanisms for adversarial learning where the discriminator works as an adversary and a critic in separate settings, enabling our model to learn expressive domain-invariant and domain-ingrained features, respectively. We collected cutting sound data from multiple sensors in different locations, prepared datasets from lab and industry domain, and evaluated our learning models on them. Experiments showed that our models outperformed the multi-layer perceptron based vanilla domain adaptation models in labeling tasks on the curated datasets, achieving near 92%, 82% and 85% accuracy respectively for three different sensors installed in industry settings.