How to to distinguish two photon Dicke/Rabi model from one photon Dicke/Rabi model in circuit QED?

I read papers about the Dicke model and Rabi model. There are many discussion about $A^2$ term in circuit QED [Gu, Xiu, et al. "Microwave photonics with superconducting quantum circuits." Physics Reports 718 (2017): 1-102.].

I heard that two photon Dicke/Rabi Hamiltonian can be transformed into one photon Dicke/Rabi Hamiltonian. Is there any proposal to distinguish two photon Dicke/Rabi model from one photon Dicke/Rabi model in circuit QED? Is there any experiment to decide the coupling constants in one photon term and two photon term in two photon Dicke/Rabi model? If there is not, why?

Also, I do not understand their motivation. Do they want to realize two photon Dicke/Rabi model or want to minimize the effect of $A^2$ term?