In quantum physics, the notion of contextuality has a variety of interpretations, which are typically associated with the names of their inventors, say, Bohr, Bell, Kochen, Specker, and recently Dzhafarov. In fact, Bohr was the first who pointed to contextuality of quantum measurements as a part of formulation of his principle of complementarity. (Instead of "contextuality," he considered dependence on "experimental conditions.") Unfortunately, the contextuality counterpart of the complementarity principle was overshadowed by the issue of incompatibility of observables. The interest for contextuality of quantum measurements rose again only in connection with the Bell inequality. The original Bohr's contextuality, as contextuality of each quantum measurement, was practically forgotten. It was highlighted in our works with applications both to physics and cognition. In this note, the theory of open quantum systems is applied to formalization of Bohr's contextuality within the framework of indirect measurements. This scheme is widely used in quantum information theory and leads to the Davis-Lewis-Ozawa theory of quantum instruments. In this scheme, Bohr's viewpoint on contextuality of quantum measurements is represented within the formal mathematical framework.