Background: Sexist and misogynistic behavior significantly hinders inclusion in technical communities like GitHub, causing developers, especially minorities, to leave due to subtle biases and microaggressions. Current moderation tools primarily rely on keyword filtering or binary classifiers, limiting their ability to detect nuanced harm effectively. Aims: This study introduces a fine-grained, multi-class classification framework that leverages instruction-tuned Large Language Models (LLMs) to identify twelve distinct categories of sexist and misogynistic comments on GitHub. Method: We utilized an instruction-tuned LLM-based framework with systematic prompt refinement across 20 iterations, evaluated on 1,440 labeled GitHub comments across twelve sexism/misogyny categories. Model performances were rigorously compared using precision, recall, F1-score, and the Matthews Correlation Coefficient (MCC). Results: Our optimized approach (GPT-4o with Prompt 19) achieved an MCC of 0.501, significantly outperforming baseline approaches. While this model had low false positives, it struggled to interpret nuanced, context-dependent sexism and misogyny reliably. Conclusion: Well-designed prompts with clear definitions and structured outputs significantly improve the accuracy and interpretability of sexism detection, enabling precise and practical moderation on developer platforms like GitHub.