The reliability of artificial intelligence (AI) in medical imaging critically depends on its robustness to heterogeneous and corrupted images acquired with diverse devices across different hospitals which is highly challenging. Therefore, this paper introduces FedAgain, a trust-based Federated Learning (Federated Learning) strategy designed to enhance robustness and generalization for automated kidney stone identification from endoscopic images. FedAgain integrates a dual trust mechanism that combines benchmark reliability and model divergence to dynamically weight client contributions, mitigating the impact of noisy or adversarial updates during aggregation. The framework enables the training of collaborative models across multiple institutions while preserving data privacy and promoting stable convergence under real-world conditions. Extensive experiments across five datasets, including two canonical benchmarks (MNIST and CIFAR-10), two private multi-institutional kidney stone datasets, and one public dataset (MyStone), demonstrate that FedAgain consistently outperforms standard Federated Learning baselines under non-identically and independently distributed (non-IID) data and corrupted-client scenarios. By maintaining diagnostic accuracy and performance stability under varying conditions, FedAgain represents a practical advance toward reliable, privacy-preserving, and clinically deployable federated AI for medical imaging.