The traditional approach to the quantum Zeno effect (QZE) and quantum anti-Zeno effect (QAZE) in open quantum systems (implicitly) assumes that the bath (environment) state returns to its original state after each instantaneous projective measurement on the system and thus ignores the cross-correlations of the bath operators between different Zeno intervals. However, this assumption is not generally true, especially for a bath with a considerably nonnegligible memory effect and for a system repeatedly projected into an initial general superposition state. We find that, in stark contrast to the result of a constant value found in the traditional approach, the scaled average decay rate in unit Zeno interval of the survival probability is generally time dependent or shows an oscillatory behavior. In the case of a strong bath correlation, the transition between the QZE and the QAZE depends sensitively on the number of measurements N. For a fixed N, a QZE region predicted by the traditional approach may in fact already be in the QAZE region. We illustrate our findings using an exactly solvable open qubit system model with a Lorentzian bath spectral density, which is directly related to realistic circuit cavity quantum electrodynamics systems. Thus the results and dynamics presented here can be verified with current superconducting circuit technology.