There is an official example that demonstrates the usage of STLExporter. I suggest you study the respective source code and use it as a foundation for your own app. You can find the mentioned example right here:

https://threejs.org/examples/misc_exporter_stl

Please note that the model I want to download is either one built from scratch in threejs or one that I imported as glb/gltb and changed some of its features in threejs.

Please keep in mind that STL has no concept of materials. It only saves the raw geometry data. STL is also unable to export hierarchical information of a scene (e.g. child-parent relationships). So depending on what you are going to change in your scene, it's likely that the respective data can't be exported as expected.

There is two implementation for export,ASCII & Binary,

        function exportASCII() {

            const result = exporter.parse( mesh );
            saveString( result, 'box.stl' );

        }

        function exportBinary() {

            const result = exporter.parse( mesh, { binary: true } );
            saveArrayBuffer( result, 'box.stl' );

        }