Recent progress in enabling new chemical reactivities by strongly coupling molecular systems to quantized radiation has stimulated theoretical developments in molecular quantum electrodynamics . In particular, light-matter interactions beyond the weak-strong coupling regime, such as the ultra-strong coupling (USC) and the deep-strong coupling (DSC) regime, are currently an active field of theoretical research . Such coupling regimes lead to new exciting physical phenomena that cannot be described with the widely used approximate light-matter Hamiltonians such as the Rabi and Jaynes-Cumming Hamiltonians of quantum optics. In this manner, it is crucial to strategically choose which light-matter Hamiltonian to use to model your system by understanding the different benefits and shortcomings of each representation.