Background: Social support (SS) and social isolation (SI) are social determinants of health (SDOH) associated with psychiatric outcomes. In electronic health records (EHRs), individual-level SS/SI is typically documented as narrative clinical notes rather than structured coded data. Natural language processing (NLP) algorithms can automate the otherwise labor-intensive process of data extraction. Data and Methods: Psychiatric encounter notes from Mount Sinai Health System (MSHS, n=300) and Weill Cornell Medicine (WCM, n=225) were annotated and established a gold standard corpus. A rule-based system (RBS) involving lexicons and a large language model (LLM) using FLAN-T5-XL were developed to identify mentions of SS and SI and their subcategories (e.g., social network, instrumental support, and loneliness). Results: For extracting SS/SI, the RBS obtained higher macro-averaged f-scores than the LLM at both MSHS (0.89 vs. 0.65) and WCM (0.85 vs. 0.82). For extracting subcategories, the RBS also outperformed the LLM at both MSHS (0.90 vs. 0.62) and WCM (0.82 vs. 0.81). Discussion and Conclusion: Unexpectedly, the RBS outperformed the LLMs across all metrics. Intensive review demonstrates that this finding is due to the divergent approach taken by the RBS and LLM. The RBS were designed and refined to follow the same specific rules as the gold standard annotations. Conversely, the LLM were more inclusive with categorization and conformed to common English-language understanding. Both approaches offer advantages and are made available open-source for future testing.