A Ballet jump takes off and lands on the same foot; a "half toe loop" takes off from one foot and lands on the other.
All toepicks, all the time.
If the skater uses a Loop BO edge to take off, then completes half a rotation and lands forward (toepick-opposite foot FI edge), it's a Falling Leaf jump. If they perform a Split jump using that entrance/landing, it's a Split Falling Leaf jump.
Both the Loop and the Half Loop take off from a BO edge and land backwards, BO / BI, respectively.
They're both considered full-rotation jumps for the simple reason that they don't land forward after a backward takeoff.