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With early critical views by Lighthill  as the starting point, a revision is performed of the classical magnetohydrodynamic (MHD) theory for the magnetized plasma properties. The importance of retaining the Hall term and using a two-fluid plasma description is stressed. The plasma Hall Magnetohydrodynamics (HMHD) description is derived by accounting for the noncentral character of the internal particle-particle forces sustaining the plasma charge neutrality. Plasma properties with similarities to the Meissner effect are predicted. The HMHD model and its extension to a current-tube plasma description are used to explain experimentally observed characteristics of magnetic confinement plasmas, with application also to cosmic plasmas.