This paper builds on efforts to adapt a game-based constitutional scheme, originally designed for Korea-like presidential systems, to semi-presidential systems, introducing two innovative schemes: Scheme A and Scheme B. Initially, we assumed the prime minister selection procedure for a vacant office could be directly transposed, but refinements led to a parallel process with relative timing, improving rigor over the original sequential, absolute-timing approach. Applying the game-based concept to the prime minister replacement procedure yields constructive vote of no-confidence. A novel concept of formal legislative confidence, grounded in a systematic theory, is then inspired, and is integrated into other procedural elements of the proposed schemes. This concept standardizes presidential dismissal authority across provisions. A ``presidential bias'' parameter shapes the schemes: Scheme A, with zero bias, mirrors a typical semi-presidential system, incorporating a Request for Successor (RFS) mechanism to unify parliamentary dissolution while minimizing its frequency. Scheme B, with a bias ranging from 0 to 1/8, adopts a presidential-leaning structure featuring a 9/16 cohabitation threshold, and excludes RFS, resulting in an indissoluble parliament. We recommend Scheme A for robust governance in fragmented parliaments and Scheme B for stable, presidential-leaning systems. As refined version, Scheme B renders the original scheme obsolete. Both proposed schemes offer innovative alternatives to traditional selection, confidence and removal mechanisms in semi-presidential systems.