AI sycophancy has become a prominent concern in large language model (LLM) research. Yet the term lacks a consistent definition and has been applied to behaviors ranging from agreeing with a user's false claim to excessively praising the user to withholding corrective feedback. When researchers, companies, and policymakers use the same term to describe different behaviors, evaluation results become difficult to compare, mitigation strategies fail to transfer, and systems that are resistant to one form of sycophancy continue exhibiting other forms. To address this, we make two contributions. First, we reviewed 70 papers on AI sycophancy to develop a taxonomy of how the behavior has been defined and measured. The taxonomy distinguishes (1) whether a model is sycophantic toward a user's positions and beliefs, or toward the user's broader personal traits and emotions, and (2) whether this occurs through explicit, direct language or more implicit, subtle behaviors such as framing, omission, or tone. Mapping existing literature to our taxonomy reveals that current research has focused on overt forms of sycophancy toward users' beliefs, leaving more subtle and person-directed behaviors relatively understudied. Second, we surveyed 106 experts in AI sycophancy and related fields to examine whether researchers agree on which model behaviors are sycophantic. While experts are nearly unanimous in believing that sycophancy is a significant problem in current AI systems (94.3% agree), they disagree substantially on which specific behaviors qualify. Together, these findings demonstrate that AI sycophancy is a broad family of behaviors with different measurement challenges, intervention requirements, and governance implications. Our taxonomy provides a shared vocabulary for understanding and addressing these behaviors.