The horizontal push trains the horizontal pressing muscles which include the chest, triceps, and anterior shoulders. This movement is important because it will teach you how to maintain a stable shoulder position while reaching out in front of you.
The horizontal pull exercise will require some piece of equipment to do it properly. You can do it with a fixed barbell, a pair of Olympic Rings, a suspension trainer, or a sturdy table. It will strengthen the muscles of the upper back and posterior shoulders.
The pike push-up will train your shoulder muscles and triceps, which will improve your overhead strength. This movement is important because it strengthens and develops your shoulders in a dynamic range of motion.
The vertical pull is an important movement because it mimics one of the most fundamental human patterns known to man- the ability to pull yourself up in any situation.