The nuclearity of C*-algebras is a fundamental property in the classification program, often viewed as an internal finite-dimensional approximation property. We introduced a refined finite-dimensional approximation property using pure states, called the Matui-Sato approximation property (MSAP). In this work, we show that this refined property is equivalent to nuclearity for separable C*-algebras. Property of such a form first appeared in the work of Matui and Sato in 2012 studying simple C*-algebras. The MSAP for simple C*-algebras plays an important role in showing the so-called property (SI), through which they made a breakthrough in the classification program: strict comparison implies Z-stability with nice enough trace space. The MSAP was later shown to hold for a large class of separable nuclear C*-algebras in 2015, and property (SI) was defined and shown for some nuclear maps. As a consequence of the equivalence between the MSAP and nuclearity, we obtain a more general property (SI) result for nuclear maps.