This paper explores uncertainty quantification (UQ) as an indicator of the trustworthiness of automated deep-learning (DL) tools in the context of white matter lesion (WML) segmentation from magnetic resonance imaging (MRI) scans of multiple sclerosis (MS) patients. Our study focuses on two principal aspects of uncertainty in structured output segmentation tasks. First, we postulate that a reliable uncertainty measure should indicate predictions likely to be incorrect with high uncertainty values. Second, we investigate the merit of quantifying uncertainty at different anatomical scales (voxel, lesion, or patient). We hypothesize that uncertainty at each scale is related to specific types of errors. Our study aims to confirm this relationship by conducting separate analyses for in-domain and out-of-domain settings. Our primary methodological contributions are (i) the development of novel measures for quantifying uncertainty at lesion and patient scales, derived from structural prediction discrepancies, and (ii) the extension of an error retention curve analysis framework to facilitate the evaluation of UQ performance at both lesion and patient scales. The results from a multi-centric MRI dataset of 444 patients demonstrate that our proposed measures more effectively capture model errors at the lesion and patient scales compared to measures that average voxel-scale uncertainty values. We provide the UQ protocols code at https://github.com/Medical-Image-Analysis-Laboratory/MS_WML_uncs.