half-loop (technically a full rotation but lands on inside edge of other foot), but you can do half of a loop, and land facing forward.
half-flip
half-lutz
Ballet jump
Mazurka
Falling Leaf
waltz jump
A bell jump is a single rotation axel, so not really a 'half' jump, but smaller than the 'normal' one.
Those are the only ones I know.
Yes, generally speaking, a "half" jump is any standard jump takeoff that is just 1/2 turn in the air and lands forward on the right toepick instead of the right back outside edge (assuming CCW jumper).
The exceptions:
Half loop - a misnomer referring to a full loop jump landed on the LBI edge instead of RBO edge. This counts as a single loop under IJS.
Waltz jump - Since you are taking off forward and doing 1/2 revolution, you land backward on the RBO edge.
Half axel (also called a bell jump) - Actually a full revolution, since you take off forward and land forward on the right toe. But it's 1/2 revolution less than a single axel.
A half jump by any other name:
Falling leaf - This is what SHOULD be called a half loop, since it's the loop jump takeoff, and landed forward on the right toe.
Mazurka - Technically, this is a half toe loop, since it takes off like a toe loop and lands forward on the right toe instead of going all the way around.
Split jump - typically done from a flip entrance, making it a half flip with a split position in the air. It can also be done from a lutz entrance, making it a half lutz with a split. A split jump done from a loop takeoff is called a split falling leaf.
Never seen:
Half salchow. Never seen one! Sounds extremely awkward. . .