A hydraulic system means the fluid is driving the system. Such a system can be either hydrostatic or hydrodynamic.
A hydrostatic system involves low-frequency actions, like the controls of a lawn mower. Pressures come to equilibrium in a fraction of a second, while the inputs are slower than that, a half-second delay in the controls just feels normal, so high-frequency effects are not important in the design.
A hydrodynamic system is more complex, and control becomes more complicated/interesting. Air-flow around a fast-moving car or airplane, or the wake of a boat, include important hydrodynamic effects, as well as hydrostatic effects. If one wrote equations describing the motion of an element of fluid, they would have both static and dynamic terms. Even engine oil has to be more concerned with this than we do, because they operate at thousands of rpm and oil splashing (why you don't overfill your crankcase) or shock waves can be a concern.
Therefore we need some sort of "plain old" hydraulic fluid, one that won't wreck our seals. "Hydrostatic fluid" should mean fluid that's designed for low-frequency applications like lawn mowers.