My information is a bit different...he was a lt. colonel in the Virginia militia at the time and not a captain. Also, he never said he fired the first shot at Jumonville Glen...since he did not.
Just to note--I may be mistaken about the rank. Washington however did sign a letter of confession to the killing. He did, iirc, so under a mistaken impression that what he was signing was an admission of a accidental killing, for his interpreter so claimed the French word for "assassin" to mean.