This is one of my favorite subjects, so I'll chime in.
The way I unerstand it, a hydrostatic system is one which is a function more of pressure than flow. Hence, "static", or stationary, with pressure being the applied force.
Hydrostatic systems, relative to hydraulic systems are generally low flow, high pressure, where components(pumps and motors) can be designed for low rpm, high torque, and use the same power source as does the hydraulic system which makes it ideal for the drive.
There are still hydraulic drive machines-like mini-excavators with fixed-displacement gear pumps and hydraulic drive motors(high rpm with planetary gear reduction).
Hydraulic systems rely more on flow than pressure. Higher flow equals faster linear motion(stroking a cylinder). Pressure is still required for digging or lifting force, but pressures are generally lower than hydrostatics sharing the power unit.
And there are load-sensing hydraulic systems that use variable-displacement pumps that are similar in design to hydrostatic pumps, which require complex controls but get you closer to the best of both systems.
I believe the future of our stuff is no more distinction between these two system types. Everything bigger than a walk-behind is going to have variable-displacement, load-sensing pumps that operate the drive as well as the work equipment with no more engine-lugging, black smoke, fuel inefficiency, and requiring engineering degrees and tall dollars to repair.