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Generative AI (genAI) tools, such as ChatGPT or Copilot, are advertised to improve developer productivity and are being integrated into software development. However, misaligned trust, skepticism, and usability concerns can impede the adoption of such tools. Research also indicates that AI can be exclusionary, failing to support diverse users adequately. One such aspect of diversity is cognitive diversity -- variations in users' cognitive styles -- that leads to divergence in perspectives and interaction styles. When an individual's cognitive style is unsupported, it creates barriers to technology adoption. Therefore, to understand how to effectively integrate genAI tools into software development, it is first important to model what factors affect developers' trust and intentions to adopt genAI tools in practice? We developed a theoretically grounded statistical model to (1) identify factors that influence developers' trust in genAI tools and (2) examine the relationship between developers' trust, cognitive styles, and their intentions to use these tools in their work. We surveyed software developers (N=238) at two major global tech organizations: GitHub Inc. and Microsoft; and employed Partial Least Squares-Structural Equation Modeling (PLS-SEM) to evaluate our model. Our findings reveal that genAI's system/output quality, functional value, and goal maintenance significantly influence developers' trust in these tools. Furthermore, developers' trust and cognitive styles influence their intentions to use these tools in their work. We offer practical suggestions for designing genAI tools for effective use and inclusive user experience.