This paper highlights the importance of clinical coding using standardized medical terminology to address the interoperability challenges of veterinary medical records, a large-scale data resource for veterinary clinical research. Compared to previous DeepTag and VetTag studies that attempted to automate veterinary diagnosis coding using LSTM and Transformer models, this study included all 7,739 SNOMED-CT diagnosis codes recognized by the Colorado State University Veterinary Teaching Hospital (CSU VTH) and fine-tuned 13 freely pre-trained language models (LMs) using 246,473 manually coded veterinary patient visit records from CSU VTH's electronic health record (EHR). The results demonstrated superior performance compared to previous studies, with the most accurate results achieved when fine-tuning a relatively large clinical LM using extensive labeled data. However, we demonstrated that similar results can be achieved even with limited resources and using non-clinical LMs. These findings contribute to improving the quality of veterinary EHRs by investigating accessible methods for automatic coding and to building an integrated and comprehensive health database spanning species and institutions to support both animal and human health research.