The parametrization of wireless channels by so-called "beyond-diagonal reconfigurable intelligent surfaces" (BD-RIS) is mathematically characterized by a matrix whose off-diagonal entries are partially or fully populated. Physically, this corresponds to tunable coupling mechanisms between the RIS elements that originate from the RIS control circuit. Here, we derive a physics-compliant diagonal representation for BD-RIS-parametrized channels. We recognize that any RIS control circuit can always be separated into its static parts (SLC) and a set of tunable individual loads (IL). Therefore, a BD-RIS-parametrized channel results from the chain cascade of three systems: i) radio environment (RE), ii) SLC, and iii) IL. RE and SLC are static non-diagonal systems whose cascade K is terminated by the tunable diagonal system IL. This physics-compliant representation in terms of K and IL is directly analogous to that for conventional ("diagonal") RIS (D-RIS). Therefore, scenarios with BD-RIS can also readily be captured by the physics-compliant coupled-dipole model PhysFad, as we show. In addition, physics-compliant algorithms for system-level optimization with D-RIS can be directly applied to scenarios with BD-RIS. We demonstrate this important implication of our conceptual finding in a case study on end-to-end channel estimation and optimization in a BD-RIS-parametrized rich-scattering environment. Our case study is the first experimentally grounded system-level optimization for BD-RIS: We obtain the characteristics of RE and IL from experimental measurements and a commercial PIN diode, respectively. Altogether, our physics-compliant diagonal representation for BD-RIS enables a paradigm shift in how practitioners in wireless communications and signal processing implement system-level optimizations for BD-RIS because it enables them to directly apply existing physics-compliant D-RIS algorithms.