As AI increasingly permeates education, understanding teachers' perceptions and interactions with these technologies is essential. The Technological Pedagogical Content Knowledge (TPACK) framework provides a comprehensive lens to examine these interactions with educational AI tools (EAITs). This study explored the factors influencing teachers' intentions to use EAITs, focusing on their TPACK profiles. It investigated how the components of TPACK--technological, pedagogical, and content knowledge--interact to shape teachers' perceptions and willingness to integrate EAITs into their teaching. Structural equation modeling analyses of quantitative data gained from 514 teachers revealed that the intention to use EAITs was positively related to the core TPACK factor, while actual use was primarily driven by the Technological Pedagogical Knowledge factor, which measures understanding of how various technologies can be applied in teaching. This study extends the TPACK framework to EAITs, offering insights into AI adoption in education. It informs professional development, guiding targeted interventions to enhance teachers' technological and pedagogical skills for effective AI integration in classrooms.