The obvious solution here is to separate the equinox into the 2 events that comprise it. The vernal/autumnal "equinoxes" happens before/after the "sun overhead at the equator" event, so acknowledge them as separate events. The equinoxes are also latitude-dependent, so everyone can have their own, personal equinox depending on their exact latitude.
The "sun overhead at the equator" could be called the "sun crossing", or the "north-south transition," or the "hemispheric transit," or something else. In keeping with Greek origins, I propose calling it the "heliodiávasi" (sun crossing). Shorten it to "heliovasi" and Bob's your uncle.