The properties of a Thyratron are high current capability (hence being able to drive the mechanical counter) and a snap-action turn on point. They are not normal tubes, they contain a gas rather like a neon lamp, and if held just below trigger voltage between the closely spaced control grid and cathode, basically do nothing. As soon a the grid-cathode space ionizes, the whole tube becomes highly conductive so your input ground should be the cathode pin. One thing that looks wrong is the placement of the capacitor C1, normally it would go from the bottom of the 300K resistor to the cathode, the idea being that it charges slowly through the resistor and dumps the charge through the counter as the tube conducts, otherwise the current would stay on when triggered and the counter would be permanently energized. The potentiometer is to pre-set the grid voltage just below trigger point so it is as sensitive as possible and any additional 'spike' from a lightning discharge pushes it over the edge into conduction.
Brian.