Birth certificates, certificates of live birth, live birth records are generally all referring to the same document. There are typically two versions of this document, whatever it may called in the state, hospital, or country where the birth takes place. The first document is the usually handwritten form completed by hospital staff at the time of birth. Its kept on file with the hospital and state department of health and vital statistics. Typically people are not given access to this official record. What you do have access to, as a matter of public record, is the formal certificate with the state seal. Its  commemorative version of the official record of ones birth and is commonly accepted as proof of citizenship and proof of live birth.

This has been my experience and I have much experience working with birth certificates due to my line of work (surrogacy/egg donation).