The United States Navy originated as the Continental Navy, established early in the American Revolution by the Continental Congress by a resolution of 13 October 1775. There is a widespread belief that ships of the Continental Navy flew a jack consisting of alternating red and white stripes, having the image of a rattlesnake stretched out across it, with the motto "Don't Tread on Me."
The Gadsden Flag
Journal of the South Carolina Provincial Congress, 9 February 1776: "Col. Gadsden presented to the Congress an elegant standard, such as is to be used by the commander in chief of the American navy; being a yellow field, with a lively representation of a rattle-snake in the middle, in the attitude of going to strike, and these words underneath, "DON'T TREAD ON ME!"
Dixon and Hunter's Virginia Gazette, Williamsburg, Virginia, 11 May 1776: "The colours of the American Fleet to have a snake with thirteen rattles, the fourteenth budding, described in the attitude of going to strike, with this motto, "Don't Tread on Me!"In early 1776 Commodore Esek Hopkins, the first and only commander in chief of the Continental Navy fleet, used a personal standard designed by Christopher Gadsden of South Carolina. This flag consisted of a yellow field with a coiled snake and the motto "Don't Tread on Me." There is no doubt as to the authenticity of Hopkins's personal standard, usually
Seems like depending on the captain, both versions were in use as a jack and also mistakenly or otherwise, referred to as "the Gadsden flag."