We present a general framework for the quasi-mutual entropy to quantify the entanglement degree for a two-level atom interacting with a Holstein-Primakoff SU(1,1) coherent state, taking into account an arbitrary form of the intensity-dependent coupling. We derive an explicit expression for the entanglement degree in terms of the density matrix elements which does not involve the diagonal approximation, and establish its general quantum theory. It is found that the quasi-mutual entropy is a good measure of the entanglement degree of the atomic state at any time. We examine the effects of the Holstein-Primakoff SU(1,1) coherent state and of the Stark shift on the entanglement degree. It is shown that features of the entanglement are influenced significantly by different values of the system parameters.