Existing saliency-guided training approaches improve model generalization by incorporating a loss term that compares the model's class activation map (CAM) for a sample's true-class ({\it i.e.}, correct-label class) against a human reference saliency map. However, prior work has ignored the false-class CAM(s), that is the model's saliency obtained for incorrect-label class. We hypothesize that in binary tasks the true and false CAMs should diverge on the important classification features identified by humans (and reflected in human saliency maps). We use this hypothesis to motivate three new saliency-guided training methods incorporating both true- and false-class model's CAM into the training strategy and a novel post-hoc tool for identifying important features. We evaluate all introduced methods on several diverse binary close-set and open-set classification tasks, including synthetic face detection, biometric presentation attack detection, and classification of anomalies in chest X-ray scans, and find that the proposed methods improve generalization capabilities of deep learning models over traditional (true-class CAM only) saliency-guided training approaches. We offer source codes and model weights\footnote{GitHub repository link removed to preserve anonymity} to support reproducible research.