Coherent two-dimensional electronic spectroscopy using incoherent (noisy) light, I((4)) 2D ES, holds intriguing challenges and opportunities. One challenge is to determine how I((4)) 2D ES compares to femtosecond 2D ES. Here, we merge the sophisticated energy-gap Hamiltonian formalism that is often used to model femtosecond 2D ES with the factorized time-correlation formalism that is needed to describe I((4)) 2D ES. The analysis reveals that in certain cases the energy-gap Hamiltonian is insufficient to model the spectroscopic technique correctly. The results using a modified energy-gap Hamiltonian show that I((4)) 2D ES can reveal detailed lineshape information, but, contrary to prior reports, does not reveal dynamics during the waiting time.