To be honest, the practical applications of the Axe kick seem limited to me. If you have to jump up in the air to try and break the clavicle, you run far too many risks. As a general rule, jumping kicks are great for learning and practising agility, speed, power and balance, but are not really so useful in the ring or on the street.
It all depends on what you want to do with the kick. An axe kick, to me is, a kick that brings a lot of force directly downwards with the heel. Useful if you can land it on top of something, like the shoulder or the head, or the guard.
A crescent kick, to me, is a kick with a horizontal force with the side of the foot. It comes up somewhat like a front kick, and then whips sideways to take out the guard, to feint, or to strike to the head. Since it can be lifted out to the side of the opponent, and since the leg does not need to be straight until the lost moment, it can be used a lot closer to the opponent. It is also a quicker, more whipping movement.
So my vote has to go to the crescent kick, for versatility and application.