This paper aims to develop an automated wound size measurement system for remote monitoring of chronic wound patients by comparing and evaluating state-of-the-art deep learning models in various aspects (general purpose vision, medical imaging, and models awarded in open wound dataset competitions). For fair comparison, we used standardized learning, data augmentation, and evaluation processes, and cross-validation was performed to minimize segmentation bias. We evaluated the generalization performance, computational efficiency, and interpretability of the models, and proposed and evaluated a reference object-based approach that converts AI-generated masks into clinically meaningful wound size estimates. Finally, we present the results of integrating the developed wound size estimation framework, WoundAmbit, into a customized remote medical care system. The Transformer-based TransNeXt model showed the highest generalization performance.