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In a Swiss birth certificate:
...Geburtsregister des Zivilstandskreises Zuerich.
Is it District Registry Office?
Also, is it correct to translate Zivilstandsbeamte as Registrar?
Your ideas and suggestions are apprecciated!
Explanation: As a general rule, you should keep the term in the original language and add a translation in brackets.
Reason is that especially public offices and rules and regulations usually have different purposes and contents than the translation suggests and may be confusing for the reader, who may get totally wrong ideas.
I think, your ideas are already good enough to go.