Fast scrambling of quantum correlations, reflected by the exponential growth of Out-of-Time-Order Correlators (OTOCs) on short pre-Ehrenfest time scales, is commonly considered as a major quantum signature of unstable dynamics in quantum systems with a classical limit. In two recent works, by Hummel et al. [1] and by Xu et al. [2], a significant difference in the scrambling rate of integrable (many-body) systems was observed, depending on the initial state being semiclassically localized around unstable fixed points or fully delocalized (infinite temperature). Specifically, the quantum Lyapunov exponent λq quantifying the OTOC growth is given, respectively, by λq=2λs or λq=λs in terms of the stability exponent λs of the hyperbolic fixed point. Here we show that a wave packet, initially localized around this fixed point, features a distinct dynamical transition between these two regions. We present an analytical semiclassical approach providing a physical picture of this phenomenon and support our findings by extensive numerical simulations in the whole parameter range of locally unstable dynamics of a Bose-Hubbard dimer. Our results suggest that the existence of this transition is a hallmark of unstable separatrix dynamics in integrable systems. This allows one to distinguish, within the exponential OTOC growth behavior, unstable integrable (many-body) dynamics from genuine chaotic dynamics featuring uniform growth.