The top method is called in-line. The side-line shown below it has equalization in the surge tank with overflow when flow rates are high into the equalization tank. This method makes sense when there is a combined sewer system for both wastewater and storm water. Much of the time the surge tank is adequate. When there are heavy rains, this overflows into the equalization tank to be proportioned into the main stream over a period of time.
The pump shown in blue is usually several pumps because one may be down for maintenance. The pumps may have constant flow, either off or on. One pump may have variable speed that is fairly expensive. The system must have some sort of logical control to decide which pumps are on and to add back the surplus wastewater at rates that keep the rest of the plant operating with nearly constant conditions.
Sludges from the various sedimentation units are digested anaerobically to reduce their organic matter and to improve handling and drying. The supernatent from treated sludge is rich in organic matter and goes back for more treatment. The equalization step is a good place to add back this supernatent.