Laser cutting, a long-established multi-physical process, has been widely adopted in the metallurgical industry, but its rapid industrialization has impacted quality control. Reviews from 2008 to 2022 primarily focus on single-criterion quality approaches, targeting defects like the Heat-Affected Zone, surface roughness, or kerf geometry, rather than adopting comprehensive methods. In addition, these studies show that cutting quality can be improved by selecting laser manufacturing parameters and part parameters such as thickness or material. However, the influence of part morphology remains under-explored. Research often limits morphology to simple segments with varying lengths or angles, neglecting a systematic analysis of its impact. To address this gap, this study evaluates the criticality of six cutting defects, as defined by existing standards, across three morphologies (arcs, segments, and angles) using an adapted Failure Modes, Effects, and Criticality Analysis method. The aim is to establish a holistic approach linking morphologies to all defect types. Industrial application reveals that thermal defects are highly influenced by morphology, with burrs and adherent slag being critical in arcs and angles. Segments, however, show less sensitivity. This analysis enables the definition of design limits and provides practical tools for improving industrial laser cutting processes through detailed quality assessments.