this works by dragging the case and moving it to the desired position in the other section (drag+drop). The icon left to the checkbox of a case is draggable. If you do not see the checkbox and the icon, please make sure that you have the Add/Edit Test Case permissions enabled for your user.
Alternatively, you can change the section of a test case by editing the test case and selecting a different section (in the box with the blue background color).
As you already found out, the Copy or Move Test Cases is only for copying/moving cases between different test suites.