This paper introduces a distributed operational solution for coordinating integrated transmission-distribution (ITD) systems regarding data privacy. To tackle the nonconvex challenges of AC optimal power flow (OPF) problems, our research proposes an enhanced version of the Augmented Lagrangian based Alternating Direction Inexact Newton method (ALADIN). This proposed framework incorporates a second-order correction strategy and convexification, thereby enhancing numerical robustness and computational efficiency. The theoretical studies demonstrate that the proposed distributed algorithm operates the ITD systems with a local quadratic convergence guarantee. Extensive simulations on various ITD configurations highlight the superior performance of our distributed approach in terms of convergence speed, computational efficiency, scalability, and adaptability.