I think the more correct term is "hydrostatic," but "hydraulic" is sometimes used as a synonym. This is not very desirable, because "hydraulic" also means other things.
Even if you test a vessel with oil (such as for vessels destined for halocarbon refrigerant service) it is still called a hydrostatic test.
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In vino veritas; in cervisia carmen; in aqua E. coli.
A hydrostatic test, as implied above, involves no flow. So it is a pressure test only. For example, a centrifugal pump rated at 16bar might be pressurized up to 24bar to ensure there are no leaks.
A hydraulic test might be better termed a performance test, so in this case the pump would be run to measure flowrate vs. pressure.
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