I would raise ValueError instead of a generic Exception

Does it matter? Is it an exception that will be caught?

It matters for the tests. The test actually catches a generic Exception and will fail to fail if the exception is raised by a coding error, whereas with a ValueError code errors can be caught. Our API users can also better understand that the exception was raised because of a value that was provided by them instead of being an error that we caused in our code.

But the tests match the exact exception message:

with pytest.raises(Exception, match="Invalid API domain"):

I would argue that setting a specific exception type will increase our API surface? Meaning that our users might start writing code that handles this exception differently. Are we sure we want to support that? Are we willing to commit to it forever?