Detecting harmful content is a crucial task in the landscape of NLP applications for Social Good, with hate speech being one of its most dangerous forms. But what do we mean by hate speech, how can we define it, and how does prompting different definitions of hate speech affect model performance? The contribution of this work is twofold. At the theoretical level, we address the ambiguity surrounding hate speech by collecting and analyzing existing definitions from the literature. We organize these definitions into a taxonomy of 14 Conceptual Elements-building blocks that capture different aspects of hate speech definitions, such as references to the target of hate (individual or groups) or of the potential consequences of it. At the experimental level, we employ the collection of definitions in a systematic zero-shot evaluation of three LLMs, on three hate speech datasets representing different types of data (synthetic, human-in-the-loop, and real-world). We find that choosing different definitions, i.e., definitions with a different degree of specificity in terms of encoded elements, impacts model performance, but this effect is not consistent across all architectures.