I wrote about the types of German birth certificate in 2005.
The practice of German register offices changed on January 1 2009.
Here is my earlier summary with the changes:
Geburtsschein (minimum details) – now gekürzte Geburtsurkunde
Geburtsurkunde (most details) – now Geburtsurkunde
Abstammungsurkunde (most details – including natural parents) – now beglaubigter Geburtsregisterausdruck
This shows that you can usually translate both Geburtsurkunde and Abstammungsurkunde as birth certificate, but there will be circumstances in which you need to distinguish them and can add ‘showing natural parents’.
Another problem I mentioned then was that some Geburtsurkunden say ‘mit der Abstammungsurkunde identisch’. This means that if you want to get married, you can use the certificate as evidence of your biological parents.
At least the translation is straightforward: we now have a short birth certificate, a (full) birth certificate (to use the UK terms) and a certified extract from the register of births.
There are more changes to the system. There is no longer a Familienbuch, but an Eheregister. I haven’t read these up in detail, but here is some information from Braunschweig.