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Abstract The images of supermassive black holes surrounded by optically thin, radiatively inefficient accretion flows, like those observed with the Event Horizon Telescope, are characterized by a bright ring of emission surrounding the black hole shadow. In the Kerr spacetime, this bright ring, when narrow, closely traces the boundary of the shadow and can, with appropriate calibration, serve as its proxy. The present paper expands the validity of this statement by considering two particular spacetime geometries: a solution to the field equations of a modified gravity theory and another that parametrically deviates from Kerr but recovers the Kerr spacetime when its deviation parameters vanish. A covariant, axisymmetric analytic model of the accretion flow based on conservation laws and spanning a broad range of plasma conditions is utilized to calculate synthetic non-Kerr black hole images, which are then analyzed and characterized. We find that in all spacetimes: (i) it is the gravitationally lensed unstable photon orbit that plays the critical role in establishing the diameter of the rings observed in black hole images, not the event horizon or the innermost stable circular orbit, (ii) bright rings in these images scale in size with, and encompass, the boundaries of the black hole shadows, even when deviating significantly from Kerr, and (iii) uncertainties in the physical properties of the accreting plasma introduce subdominant corrections to the relation between the diameter of the image and the diameter of the black hole shadow. These results provide important new theoretical justification for using black hole images to probe and test the spacetimes of supermassive black holes.