Do you think that there should be a different name for the cycle if a player hits the cycle but in order. single, double, triple, homerun? Maybe call it a king cycle or something?

There is, it is called the "Natural cycle"

so this is the same idea as a "hat trick" in hockey, but to be a TRUE hat trick, the player has to score three goals IN A ROW, not just three a game.

I t's called the natural cycle.