Effective representation of medical concepts is crucial for secondary analyses of electronic health records. Neural language models have shown promise in automatically deriving medical concept representations from clinical data. However, the comparative performance of different language models for creating these empirical representations, and the extent to which they encode medical semantics, has not been extensively studied. This study aims to address this gap by evaluating the effectiveness of three popular language models - word2vec, fastText, and GloVe - in creating medical concept embeddings that capture their semantic meaning. By using a large dataset of digital health records, we created patient trajectories and used them to train the language models. We then assessed the ability of the learned embeddings to encode semantics through an explicit comparison with biomedical terminologies, and implicitly by predicting patient outcomes and trajectories with different levels of available information. Our qualitative analysis shows that empirical clusters of embeddings learned by fastText exhibit the highest similarity with theoretical clustering patterns obtained from biomedical terminologies, with a similarity score between empirical and theoretical clusters of 0.88, 0.80, and 0.92 for diagnosis, procedure, and medication codes, respectively. Conversely, for outcome prediction, word2vec and GloVe tend to outperform fastText, with the former achieving AUROC as high as 0.78, 0.62, and 0.85 for length-of-stay, readmission, and mortality prediction, respectively. In predicting medical codes in patient trajectories, GloVe achieves the highest performance for diagnosis and medication codes (AUPRC of 0.45 and of 0.81, respectively) at the highest level of the semantic hierarchy, while fastText outperforms the other models for procedure codes (AUPRC of 0.66). Our study demonstrates that subword information is crucial for learning medical concept representations, but global embedding vectors are better suited for more high-level downstream tasks, such as trajectory prediction. Thus, these models can be harnessed to learn representations that convey clinical meaning, and our insights highlight the potential of using machine learning techniques to semantically encode medical data.