Reading your link, it looks like by "hydrostatic burning", they mean the nuclear fusion that occurs while the star is still in hydrostatic equilibrium. People often refer to nuclear fusion reactions as"burning", by analogy with chemical burning. Any normal star is in hydrostatic equilibrium, with the inward pressure due to gravity being balanced by the radiation pressure from the energy released due to nuclear fusion. Note that this hydrostatic burning phase lasts for billions of years for our sun, and millions of years for a star of 10 solar masses. The hydrostatic burning phase occurs before the core collapse that triggers a core collapse supernova.