Five British officers who were on the green when the shooting started recorded their observations -- Major John Pitcairn, Lieutenant William Sutherland, Ensign Jeremy Lister, Lieutenant John Barker, and Lieutenant Edward Thoroton Gould. Of these, Pitcairn, Sutherland, and Lister clearly asserted how the firing began.
Recently, I stumbled upon a sixth account by a British officer -- Captain William Glanville Evelyn -- who led the light infantry company of the 4th Regiment of Foot. Evelyn's account appears in a letter dated April 23, 1775, and addressed to his father. He wrote:
On the night of the 18th instant, the Grenadiers and Light Infantry of our little army, making near 700 men, embarked privately, and crossed above the common ferry here, in order to go to a town about twenty miles off, to destroy some cannon, provision, &c., that had been collected there; the country having been alarmed by the appearance of troops in the night, they assembled from every quarter; and within about five miles of the place (Concord), our men found themselves opposed by a body of men in arms, whose design appeared to be to stop their progress. This they were soon convinced of, by receiving a scattering shot or two from them, upon which a few of our people fired, and killed seven or eight minute men; and so passed on to Concord, where they destroyed some iron guns, gun-carriage wheels, and about 100 casks of flour.
In summarizing the evidence about the events on Lexington Green, I wrote, “there is a measure of agreement that either one shot, or a few shots occurring in close succession, immediately preceded a volley by the British regulars.” In this respect, Evelyn's account matches that of other participants and eyewitnesses. However, Evelyn's description does not exactly match that of the other British officers, and it certainly differs from that of those American participants who flatly stated the British fired first.
For the rest of Captain Evelyn's description of the battle of Lexington and Concord, and other details of his Revolutionary War service, see here.