In this work, we have provided a theoretical and numerical comparison of two complementary approaches to ai-QED: (1) parameterized CQED (pQED), a twostep approach where the matter degrees of freedom are computed using existing electronic structure theories, enabling one to build rigorous ai-QED Hamiltonians in a basis of many-electron eigenstates, and (2) self-consistent CQED (scQED), a one-step approach where electronic structure methods are generalized to include coupling between electrons and photon degrees of freedom. Using simple theoretical arguments, we have identified a disparity between the projection of the two-body dipole self energy operator that appears in the pQED approach and its exact operator counterpart in the scQED approach. We provided a simple theoretical argument that this disparity resolves only under the limit of a complete orbital basis and in the limit of a complete many-electron basis for the projection. We provided numerical results highlighting this disparity and its resolution on simple molecular systems where it is possible to approach these two complete basis limits simultaneously. We also examined and compared the practical issue of computational cost to converge each approach towards the complete orbital and many-electron basis, suggesting that in for light-matter coupling strengths that require a large number of electronic states to converge pQED approaches, directly solving for coupled states in scQED approaches will likely be more computationally efficacious. We also examined several situations that highlight that pQED and scQED can show remarkably similar convergence behavior with respect to the photonic Fock space. These findings should prove extremely useful context for the further development and selection of both approaches to strong lightmatter coupling.