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)
Sources: