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Explanation: I'm not quite sure what you're asking - the Geburtenbuch is kept in the Standesamt (is that what you meant by "The note that stays with the registrar when you register the birth of a child"?).
The birth is registered (wird ins Geburtenbuch eingetragen), the registrar issues the birth certificate (Geburtsurkunde). Note that there is a difference:
Abschrift aus dem Geburtenbuch. Im Gegensatz zur Geburtsurkunde, die nur einen Auszug
aus dem Geburtenbuch einer Personenstandsbehörde darstellt und die nur ... www.help.gv.at/99/990002_f.html