  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.
