Real-time segmentation of the liver parenchyma is a key requirement for computer-assisted guidance in minimally invasive liver surgery. However, existing methods are developed mainly on public cholecystectomy datasets and are therefore poorly representative of institution-specific acquisition conditions and of the appearance changes introduced by active liver resection. This study investigates public-to-private domain shift in real-time liver segmentation and proposes a SegFormer-B0-based boundary-guided framework that leverages anatomical liver contours and resection-induced boundaries to refine parenchyma segmentation. A surgery-specific augmentation strategy complements the framework by simulating intraoperative perturbations and generating synthetic resection-margin supervision when corresponding annotations are unavailable in the public training data. The method was evaluated on public laparoscopic data and private robot-assisted liver surgery data including both normal anatomy and active resection scenes. The proposed framework consistently outperformed the plain SegFormer-B0 baseline under both Public Training and combined Public + Private Training configurations. In the Public Training setting, the largest improvement was observed on NKI-Resection, where Dice increased from 0.71 to 0.87, while maintaining real-time inference at 75.50 FPS. A fixed library of ten manually annotated private resection images, drawn exclusively from the private training procedures, was used as a source of augmentation patches. Ablation results indicated complementary contributions from surgery-specific augmentation and resection-margin boundary modeling. These findings support boundary-guided refinement and targeted surgical augmentation as promising strategies for improving real-time liver segmentation under domain shift.