Why is this chart called a Short Field Effort Takeoff Ground Roll and not simply a Short Field Takeoff Ground Roll?
I think it is used in the same meaning as the maximum effort takeoff/landing (similar to @mins comment above). Maximum effort takeoff is used to reduce the ground roll/obstacle clearance distance. Maximum effort landing (or assault landing, as it is called) is used to reduce the ground roll after landing. This is mostly used for military aircraft, which may have to operate from small, improvised forward air strips.

This is alluded to in the PA-44-180 manual:


  In certain short field takeoff efforts when the shortest possible ground roll and the greatest clearance distance over a 50 ft. obstacle is desired, a flap setting of 25° is recommended.


This can also be figured out from the charts themselves. For Pressure altitude of 680', OAT of 8$^{\circ}C$, 8kt headwind and 3430 lb TOW, the normal TO roll is 920', while the short filed effort TO roll is only 800'. For the same specifications, the total distance over a 50' barrier using normal procedures is 1750', while using short field effort, it reduces to 1280' (All figures from the manual).

You can see a C130J demonstrating max-effort TO and landing here.