The 'fumi komi' or crossover kick is another inside crescent kick, but applied to a different target with different focus. Many do not realize that it's a crescent kick, event those who perform it.
Do you happen to have something showing this in use? The way it's shown in these videos, I'd consider it more of a stomp kick than a crescent kick. For it to count as what I'd call a crescent kick, the impact needs to be with the edge of the foot (inside edge, for the movement shown) while it's moving more or less parallel to the ground.