Yeah, all has changed within the last few years...
from a recruiting pov: I don't need a real SQL Server DBA anymore, as everything is in the cloud and managed (most of it) by the provider... for my business needs I just need some people who can handle data => data scientists or data engineer... and if there - at some point - is the need for a DBA, my data xxx people can handle that.
So that's why you can find a "SQL Server DBA" job anymore...
regarding your question about which role contains DBAs... unfortunately none really... DP-300 Azure Database Administrator contains most of the parts a former DBA actually needs to know but is more a cloud data architect role than a SQL Server DBA... so if you want to start your Cloud Data journey... start with DP-900 - Azure Data Fundamentals and then learn for DP-300 (or take a look into data scientist/engineer)
And for the second part of your question... yes, everything was "renamed" or changed over the years => earlier you run any ETL processing with an SSIS package on an SSIS server... there are still a lot of SSIS servers somewhere but as everyone migrates to the cloud (Azure)... there it is called Azure Data Factory where you create your workflows and those workflows using data pipelines, formerly known as DWH (on-premise) is now Azure Synapse. and so on and so on...
You might get more familiar with all those new names if you start your new journey with Azure Data Fundamentals (DP-900)
https://learn.microsoft.com/en-us/learn/certifications/azure-data-fundamentals/