Trust is one of the most important factors shaping whether and how people adopt and rely on artificial intelligence (AI). Yet most existing studies measure trust in terms of functionality, focusing on whether a system is reliable, accurate, or easy to use, while giving less attention to the social and emotional dimensions that are increasingly relevant for today's generative AI (GenAI) systems. These systems do not just process information; they converse, respond, and collaborate with users, blurring the line between tool and partner. In this study, we introduce and validate the Human-AI Trust Scale (HAITS), a new measure designed to capture both the rational and relational aspects of trust in GenAI. Drawing on prior trust theories, qualitative interviews, and two waves of large-scale surveys in China and the United States, we used exploratory (n = 1,546) and confirmatory (n = 1,426) factor analyses to identify four key dimensions of trust: Affective Trust, Competence Trust, Benevolence & Integrity, and Perceived Risk. We then applied latent profile analysis to classify users into six distinct trust profiles, revealing meaningful differences in how affective-competence trust and trust-distrust frameworks coexist across individuals and cultures. Our findings offer a validated, culturally sensitive tool for measuring trust in GenAI and provide new insight into how trust evolves in human-AI interaction. By integrating instrumental and relational perspectives of trust, this work lays the foundation for more nuanced research and design of trustworthy AI systems.