Actually Maxwell introduced the concept of displacement current. And with this modification the system became complete. He certainly based his work on Gauss, Faraday, and Ampere. But before he did his modification to Ampere's law, the only theoretical way to produce a magnetic field was to have a current (charges in motion). He added the notion that a time varing electric field also produced a magnetic field. He set up an experiement where a constant current was charging a capacitor and noted that a magnetic field existed in between the plates of the cap where there was clearly no current flowing. He also noted that this magnetic field had the same total flux as the field surounding the conductors leading to and from the capacitor.
While Einstein used the Lorentz transform, he changed the reasoning behind its usage and its actual interpretation. However people still call the Lorentz transform a "Lorentz" transform. Einstein's greatest work was in GR. SR is pretty easy to follow, but Einstein had the forthought to choose the proper two axioms (velocity of light is constant in vacuo and physics is the same everywhere) and show that Maxwell's wave equation is invariant under Lorentz transformation. GR took a lot more vision and the use of tensors. The biggest contributor would be Poincare'.
His work on GR cost him 11 years of work after SR and his marriage. Other scientists warned him to not work on gravity since it is too hard. Not to take away from Hilbert's mathematical ability, I never seen his work mentioned in relativity books saying that he predated Einstein.