Binary-outcome measurements allow to determine whether a multi-level quantum system is in a certain state while preserving quantum coherence between all orthogonal states. In this paper, we explore different regimes of the dispersive readout of a three-level superconducting quantum system coupled to a microwave cavity in order to implement binary-outcome measurements. By designing identical cavity frequency shifts for the first and second excited states of the system, we realize strong projective binary-outcome measurements onto its ground state with a fidelity of $94.3%$. Complemented with standard microwave control and low-noise parametric amplification, this scheme enables the quantum non-demolition detection of leakage errors and can be used to create sets of compatible measurements to reveal the contextual nature of superconducting circuits.
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