Static event-triggering-based control problems have been investigated when implementing adaptive dynamic programming algorithms. The related triggering rules are only current state-dependent without considering previous values. This motivates our improvements. This article aims to provide an explicit formulation for dynamic event-triggering that guarantees asymptotic stability of the event-sampled nonzero-sum differential game system and desirable approximation of critic neural networks. This article first deduces the static triggering rule by processing the coupling terms of Hamilton-Jacobi equations, and then, Zeno-free behavior is realized by devising an exponential term. Subsequently, a novel dynamic-triggering rule is devised into the adaptive learning stage by defining a dynamic variable, which is mathematically characterized by a first-order filter. Moreover, mathematical proofs illustrate the system stability and the weight convergence. Theoretical analysis reveals the characteristics of dynamic rule and its relations with the static rules. Finally, a numerical example is presented to substantiate the established claims. The comparative simulation results confirm that both static and dynamic strategies can reduce the communication that arises in the control loops, while the latter undertakes less communication burden due to fewer triggered events.