You have illustrated the wrong document. This document is for a freed slave. Well spotted, but beside the point. I searched for "seaman protection papers" and that was the best I could do. In fact, there were many thousands of them created, and they were used to convince the British that the person being interviewed was an American citizen.
The point remains. Papers to prove you are a citizen did not begin in 1915. They were at least a century earlier. 
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I had also thought to use the case of James McClure who presented his proof of being born in Charleston South Carolina to John Armstrong, (Ambassador to France), but John Armstrong (an actual Constitutional Delegate) refused to accept them as proof he was a citizen.
British officers actually acted with some care when impressing seamen as they could be held criminally liable for impressing someone not eligible.
Many months ago when I was researching this in more detail, I believe I read that many natural born American citizens were also detained and impressed. Thousands, in fact. The British simply asserted the rule "Born to a British Father makes you British."
Interesting stuff, but nothing whatever to do with birth certificates and eligibility for elected office.
Evidence that paper proof of citizenship was asserted and understood during the 1790s.
Just a few points of clarification to your comments. The discussion was about birth certificates, not seaman protection papers or manumission certificates. These were not birth certificates per se and were only held by a small fraction of the population. Birth certificates did not come into general usage until after 1900.
Mr. Madison’s comments were made after he became President in 1809 and were, in part, in response to the failure of seamen protection papers to prevent impressment of American sailors and of the aftermath of the Chesapeake-Leopard affairs which was an armed assault on American sovereignty. Commodore Barron of the Chesapeake was tried by Court Martial for his failure to defend his ship. The United States refused to accept the British demand that a sailor found without such papers could be legally pressed, we argued that their presence on an American vessel was sufficient to assert their status as Americans. In other words, they didn’t need no Stinkin’ Papers.
The numbers of Americans caught up in the press is not known, but certainly at least 1000. They fall into several categories:
1. Americans who had freely joined the Royal Navy (usually after a voyage to England), subsequently deserted and were really being returned to their enrolled condition rather than impressed.
2. British subjects who had immigrated to America and become naturalized citizens, a status not recognized by Britain.
3. American citizens born on U.S. soil. There were not too many of these as the laws of impressment did not permit it, but it did occur either in cases of mistaken identity or through arrogance of the Royal Navy officers involved.