This study examines the cognitive strategies users employ when interacting with a conversational AI system across three experimentally assigned tasks: creative writing, concept explanation, and text proofreading. Using an LLM-based classification pipeline, 524 US participants were coded along eight psychological dimensions grounded in motivation, self-regulation, and epistemic cognition theory. Latent class analysis identified six distinct cognitive strategy profiles that predict engagement depth more strongly than any individual dimension. A two-stage hurdle model reveals that engagement operates through separate mechanisms: self-regulation strategy determines whether users engage beyond a single turn, while need for cognition drives depth among those who do. Task difficulty did not differ across conditions and thus cannot account for the observed strategy differences. The findings offer a psychologically grounded typology of user engagement with conversational AI and demonstrate the feasibility of LLM-based conversation classification at scale.