Your understanding of a real caustic (I presume you call it real as opposed to the imaginary caustic that you mention later) is correct.
First the easy part: an imaginary caustic is a caustic located on the extension of the light rays beyond the optical system from which they arrive. For instance, in the presence of a convex lens, imaginary caustics may form behind the lens itself.
Then the more difficult part. Geometrical optics can be developed under the assumption that the wavelength (for a monochromatic wave) $\lambda$ is much smaller than any other physical length in the process. In this case, all physical quantities can be shown to be proportional to
$$f \propto e^{\imath\psi/\lambda}$$
where the function $\psi$ is the wave phase. Surfaces with $\psi =$ constant are wave surfaces, and caustics are the regions (if any) which are reached by (at least) two distinct wave surfaces, say one with $\psi = \psi_1$ and one with $\psi=\psi_2$. This mathematical characterization is global, not local, because the bending of light rays may occur either locally or remotely, and then lead to photons crossing paths.
There is another characterization of caustics, by means of the angular eikonal: if you are interested, you can find it in Landau & Lifshitz's Theory of Fields. In that case, finding caustics is shown to be equivalent to finding multiple solutions a system of four equations.
Lastly, you are right that near caustics geometrical optics breaks down; as a matter of fact, as rays cross caustics, there is this cute little phenomenon whereby their phase is changed by exactly $-\pi/2$, which is of course inexplicable in geometrical optics. Again, see Landau and Lifshitz for a poignant explanation.