We have studied the dynamic phase transition (DPT) of the kinetic Ising model in systems with surfaces within the mean-field approximation. Varying the surface exchange coupling strength J(s), the amplitude of the externally applied oscillating field h(0), and its period P, we explore the dynamic behavior of the layer-dependent magnetization and the associated DPTs. The surface phase diagram shows several features that resemble those of the equilibrium case, with an extraordinary bulk transition and a surface transition for high J(s) values, independent from the value of h(0). For low J(s), however, h(0) is found to be a crucial parameter that leads to nonuniversal surface behavior at the ordinary bulk transition point. Specifically, we observed here a bulk-supported surface DPT for high field amplitudes h(0) and correspondingly short critical periods P(c), whereas this surface transition simultaneous to the bulk one is suppressed for slow critical dynamics occurring for low values of h(0). The suppression of the DPT for low h(0) not only occurs for the topmost surface layer, but also affects a significant number of subsurface layers. We find that the key physical quantity that explains this nonuniversal behavior is the time correlation between the dynamic surface and bulk magnetizations at the bulk critical point. This time correlation has to pass a threshold value to trigger a bulk-induced DPT in the surface layers. Otherwise, dynamic phase transitions are absent at the surface in stark contrast to the equilibrium behavior of the corresponding thermodynamic Ising model. Also, we have analyzed the penetration depth of the dynamically ordered phase for the surface DPT that occurs for large J(s) values. Here we find that the penetration depth depends strongly on J(s) and behaves identically to the corresponding equilibrium Ising model.