Video summarization helps turn long videos into clear, concise representations that are easier to review, document, and analyze, especially in high-stakes domains like surgical training. Prior work has progressed from using basic visual features like color, motion, and structural changes to using pre-trained vision-language models that can better understand what's happening in the video (semantics) and capture temporal flow, resulting in more context-aware video summarization. We propose a three-stage framework, PRISM: Procedural Representation via Integrated Semantic and Multimodal analysis, that produces semantically grounded video summaries. PRISM combines adaptive visual sampling, label-driven keyframe anchoring, and contextual validation using a large language model (LLM). Our method ensures that selected frames reflect meaningful and procedural transitions while filtering out generic or hallucinated content, resulting in contextually coherent summaries across both domain-specific and instructional videos. We evaluate our method on instructional and activity datasets, using reference summaries for instructional videos. Despite sampling fewer than 5% of the original frames, our summaries retain 84% semantic content while improving over baselines by as much as 33%. Our approach generalizes across procedural and domain-specific video tasks, achieving strong performance with both semantic alignment and precision.