Two applications of the Noether method for fluids and plasmas are presented based on the Euler-Lagrange and Euler-Poincaré variational principles, which depend on whether the dynamical fields are to be varied independently or not, respectively. The relativistic cold laser-plasma equations, describing the interaction between an intense laser field with a cold relativistic electron plasma, provide a useful set of equations amenable to both variational formulations. The derivation of conservation laws by the Noether method proceeds from the Noether equation, whose form depends on the variational formulation used. As expected, the expressions for the energy-momentum conservation laws are identical in both variational formulations. The connection between the two Lagrangian densities is shown to involve the mass conservation and Lin constraints associated with the cold relativistic electron fluid.