Learner-item cognitive modeling plays a central role in the web-based online intelligent education system by enabling cognitive diagnosis (CD) across diverse online educational scenarios. Although ID embedding remains the mainstream approach in cognitive modeling due to its effectiveness and flexibility, recent advances in language models (LMs) have introduced new possibilities for incorporating rich semantic representations to enhance CD performance. This highlights the need for a comprehensive analysis of how LMs enhance embeddings through semantic integration across mainstream CD tasks. This paper identifies two key challenges in fully leveraging LMs in existing work: Misalignment between the training objectives of LMs and CD models creates a distribution gap in feature spaces; A unified framework is essential for integrating textual embeddings across varied CD tasks while preserving the strengths of existing cognitive modeling paradigms to ensure the robustness of embedding enhancement. To address these challenges, this paper introduces EduEmbed, a unified embedding enhancement framework that leverages fine-tuned LMs to enrich learner-item cognitive modeling across diverse CD tasks. EduEmbed operates in two stages. In the first stage, we fine-tune LMs based on role-specific representations and an interaction diagnoser to bridge the semantic gap of CD models. In the second stage, we employ a textual adapter to extract task-relevant semantics and integrate them with existing modeling paradigms to improve generalization. We evaluate the proposed framework on four CD tasks and computerized adaptive testing (CAT) task, achieving robust performance. Further analysis reveals the impact of semantic information across diverse tasks, offering key insights for future research on the application of LMs in CD for online intelligent education systems.