Multi-energy coupled low-carbon distribution networks (MEC-LCDNs) face growing risks from extreme weather and high-order contingencies. Traditional deterministic criteria (e.g., N-1) often overlook these low-probability, high-impact events, while existing simulation-based probabilistic methods suffer from excessive computational burdens and a lack of intuitive visualization. To address these challenges, this paper proposes an explicit representation method for MEC-LCDN operational reliability constraints based on the probabilistic reliability region (PRR). This approach transforms the abstract probabilistic reliability criterion—loss of load probability (LOLP)—into a visualizable geometric space. Specifically, a fast contingency screening technique (FCST) is developed to identify a minimal set of boundary scenarios that anchor the target reliability threshold. Subsequently, complex probabilistic constraints are decoupled into deterministic N-k security constraints under these boundary scenarios, enabling the analytical construction of the PRR boundary. A case study demonstrates that the proposed method reduces the number of required contingency scenarios by over 90% and slashes computation time from 78.8 s to 3.1 s compared to traditional N-k truncation methods. Furthermore, the method accurately quantifies the system’s total supply capability (TSC) at 44.501 MW while providing intuitive visualizations of reliability boundaries that satisfy stringent LOLP criterion.