As large language model (LLM) chatbots become embedded in education, researchers need scalable ways to analyze what learners are actually doing in student-chatbot interactions. To address this challenge, we propose a learning theory-informed annotation framework for learner intent and engagement in educational chatbot dialogues, operationalizing solution seeking, knowledge seeking, cognitive engagement, and mental model expression through a hierarchical decision process, and apply it manually to a dataset of 2,393 learner-chatbot interactions (408 double-coded). We then evaluate two LLMs as zero-shot annotators and compare human-human, LLM-LLM, and human-LLM agreement across both intermediate decisions and final intent labels. While the two LLMs exhibit agreement patterns approaching human-human reliability on the final label (human κ = 0.73 vs. LLM κ = 0.67), their agreement with human reference labels is substantially lower (κ ≤ 0.53), indicating that automated annotation errors cannot be understood from aggregate scores alone. Through exploratory qualitative coding of disagreement cases, we find that divergences can arise from ambiguous learner intent, competing valid interpretations, and human annotation variability, rather than model failure alone.