Yes, that is correct.

Since 2020-01-01, the former **Berliner Ausländerbehörde** is called **Landesamt für Einwanderung**. It often takes time for such name changes to be used consistently and often even longer before many becomes aware or used to the name change. 

Since most towns (or county (**Kreis**)) in Germany have one for their areas of responsiblity, the name will often include the city, county name. 

Since the organization is done per state, each state may have a different name for the authority:

- Ausländerbehörde
- Ausländeramt
- Zentrale Ausländerbehörde
- Fremdenamt/polizei/wesen (Austria) 
- Migration (Switzerland) 

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**Sources**:

- [Ausländerbehörde – Wikipedia][auslaenderbehoerde_wiki_de] 
- [Namenswechsel: Berliner Ausländerbehörde ist jetzt Einwanderungsamt | rbb24][auslaenderbehoerde_rbb_de] 

[auslaenderbehoerde_wiki_de]:https://de.m.wikipedia.org/wiki/Ausl%C3%A4nderbeh%C3%B6rde
[auslaenderbehoerde_rbb_de]:https://www.rbb24.de/politik/beitrag/2020/01/umbenennung-auslaenderbehoerde-landesamt-einwanderung-berlin.html