A child who is born outside of the US to American parents receives a birth certificate from the State Department “Certificate of Birth Abroad of a Citizen of the United States of America.”
My son has one. He is as fully American as his older brother who was born in Texas.
The Form FS-240 is not a birth certificate, such as is issued by a government-authorized bureau or office of vital statistics, because consular officers are not authorized to assume a foreign local or state vital statistics function.