Answering my own question here, hoping someone else might find this useful:
I was under the (false) impression that the unique "DataSourceID" given to a data source on the server would be sufficient to identify it uniquely.
So in my generated RDL, I had something like :
<DataSources>
<DataSource Name="MyDataSource">
<Transaction>true</Transaction>
<DataSourceReference>MyDataSource</DataSourceReference>
<rd:DataSourceID>6ba7c588-e270-4de9-988c-d2af024f10e1</rd:DataSourceID>
<rd:SecurityType>None</rd:SecurityType>
</DataSource>
</DataSources>
Now this worked once, when my data source was indeed called "MyDataSource" and located in the same directory as my report which I published through the RS WebService API.
As soon as I moved the data source elsewhere, it stopped working.
THE SOLUTION:
This may sound silly, but I really didn't "get it" at first: the DataSourceReference
needs to have the full and complete "path" on the Reporting Server to that data source I want to reference. Just specifying the unique ID won't do....
So once I changed my RDL to:
<DataSources>
<DataSource Name="MyDataSource">
<Transaction>true</Transaction>
<DataSourceReference>/MyProject/DataSources/MyDataSource</DataSourceReference>
<rd:DataSourceID>6ba7c588-e270-4de9-988c-d2af024f10e1</rd:DataSourceID>
<rd:SecurityType>None</rd:SecurityType>
</DataSource>
</DataSources>
(notice the <DataSourceReference>/MyProject/DataSources/MyDataSource</DataSourceReference>
)
since that moment it works like a charm.
Hope someone might find this useful some day!
Best Answer
Here is how I change a data source at runtime.
Create another "dsDynamic" data source and set its connection string expression to the following:
="Data Source="+Parameters!DatabaseServer.Value+";Initial Catalog=" + Parameters!DatabaseName.Value
Build and test the report by setting all datasets to the test connection created in step 3.