This paper introduces and systematically explores the notion of deferred $\mathcal{I}$-lacunary statistical convergence and strongly deferred $\mathcal{I}$-lacunary Cesàro convergence for sequences of real numbers, unifying deferred intervals, lacunary sequences and ideal convergence. We establish their fundamental properties and relationships between them. Furthermore, we define and investigate the corresponding deferred $\mathcal{I}$-lacunary statistical limit superior and limit inferior, providing characterizations for bounded and convergent sequences. The framework is extended to the power series method. As a key application, we establish a Korovkin-type approximation theorem using a newly introduced convergence framework based on the power series method, demonstrating its broader applicability than the classical version.
