Most of the lawn tractors use a vertical shaft engine that belt drives the trans-axle UNIT. Inside that trans-axle unit is a hydrostatic pump that pushes oil into a drive motor which in turn, applies rotary power to the differential gears.
These are self-contained units. There are some garden tractors with horizontal shaft engines that use a shaft to apply power to the hydrostatic trans-axle.
Hydrostatic pumps also have what is called a "charge pump" that keeps a constant flow of oil passing through the hydro pump to keep it full of oil as well as to provide cooler oil. On certain models of hydro trans-axles, there are ports leading to the charge pump and those ports can be used to provide hydraulic power for cylinders to raise/lower attachments, make a snow blade pivot right and left or allow power steering to be added.
You normally do not see this feature on lawn tractors but it is not unusual to find it on garden tractors.
Whether a belt or a drive shaft is used to connect the engine to the trans-axle, both are spinning the hydro-static pump.
Hopefully, the above will clear up your confusion.