A strong leg drive to start is used in both lifts.
A push press is after the leg drive, you finish with a pressing of the shoulders. Your feet don't leave the start position and your legs are straight.
A push jerk is after the leg drive, you have to catch the bar with the arms locked. There is no pressing of the shoulders. In order to meet the bar with locked arms, you will have to dip your legs a bit. If you have to dip deep to catch or ride the weight into a squat, it is a squat jerk.
A split jerk is after the leg drive, you split your legs to catch the bar in the overhead position with straight arms. No pressing out of the bar is allowed.