This study reconciles discrepancies between practical electrochemical conditions and theoretical density functional theory (DFT) frameworks, evaluating three graphyne-confined single-atom catalysts (Mo-TEB, Mo@GY, and Mo@GDY). Using both constant charge models in vacuum and constant potential models with continuum implicit solvation, we closely mimic real-world electrochemical environments. Our findings highlight the crucial role of explicitly incorporating electrode potential and pH in the constant potential model, providing enhanced insights into the nitrogen reduction reaction (NRR) mechanisms. Notably, the superior NRR performance of Mo-TEB is attributed to the d-band center's proximity to the Fermi level and enhanced magnetic moments at the atomic center. This research advances our understanding of graphyne-confined single-atom catalysts as effective NRR platforms and underscores the significance of the constant potential model for accurate DFT studies of electrochemical reactions.