A manufacturing method is provided. During this method, a preform component is provided for a turbine engine. The preform component includes a substrate comprising electrically conductive material having an outer coating comprising non-electrically conductive material applied over a surface of the substrate. A preform aperture is formed in the preform component using an electrical discharge machining electrode. The preform aperture includes a meter section of a cooling aperture in the substrate. The preform aperture also includes a pilot hole in the outer coating. A diffuser section of the cooling aperture is formed in at least the outer coating using a second machining process.