Good question++. Your second pared-down version of the module just adds one more sub to the main package. Try adding the line
at the top of the second version, and see what happens.
Your question begins to tease out three Perl facilities that are so often used together that it's easy to miss that they are distinct. One is the ability to define namespaces using the package keyword. The other is the ability to load source from a different file (e.g. with use). The third one is the ability to export/import names to/from one namespace to another (facilitated by Exporter).