As far as I know, the first completely correct published proof of the main theorem in the Artin-Verdier seminar (Theorem 3.1) is that in the second edition of my book on Arithmetic Duality Theorems.
Voloch has posted more complete information on the Lubin, Serre, Tate seminar here
For a beautiful report on the conference, see Serre's letter to Grothendieck, August 2-3, 1964 (Grothendieck-Serre Correspondence, AMS 2004).
The following summer [of 1964], shortly after his election to the National Academy of Sciences, [Zariski] organized a conference on algebraic geometry at Woods Hole, Massachussetts. Like the summer institute at Boulder ten years before, it was devoted to a specific topic and sponsored by the American Mathematical Society. It marked the beginning of the AMS sponsorship of what has become a tradition of summer institutes on a specific area of mathematics.
A seminal conference, it brought algebraic geometers from all over the world together for the first time during a very exciting period. Among Zariski's own students, Hironaka had recently proved new results on resolution, Mumford had results on the theory of moduli, and Artin was finding interesting results on étale cohomology.
The table of contents of the proceedings demonstrates how thoroughly Zariski's influence pervaded what was still, in those days, a very small field. Abhyankar, Hironaka, and Zariski gave all three of the papers presented on the theory of singularities. Of the six lectures on the classification of surfaces and moduli, two are attributed to Zariski's students, Mumford and Rosenlicht, and two to his protégés, Igusa and Nagata; and of the three papers on Grothendieck cohomology, one is by Artin and one is by Zariski's Harvard colleague, John Tate.
Many major problems would be solved during the next decade using the tools that Zariski and Weil and Grothendieck had developed for algebraic geometry. "The summer institute gave us the idea that algebraic geometry was now a real subject, and no longer simply a mass of iffy results with a few valiant people like Zariski and Weil struggling to make order in this Augean Stable." (Mumford)