Most of the time you need to create a Service to perform something in the background, and your visible Activity simply controls this Service. (I'm sure the Music player works in the same way, so the example in the docs seems a bit misleading.) If that's the case, then your Activity can finish as usual and the Service will still be running.
A simpler approach is to capture the Back button press and call moveTaskToBack(true) as follows:
// 2.0 and above
@Override
public void onBackPressed() {
moveTaskToBack(true);
}
// Before 2.0
@Override
public boolean onKeyDown(int keyCode, KeyEvent event) {
if (keyCode == KeyEvent.KEYCODE_BACK) {
moveTaskToBack(true);
return true;
}
return super.onKeyDown(keyCode, event);
}