In the picture I've selected the edge loop I would like to see "cut out" like a bevel, so like a kind of groove in the cube's surface. The Bevel Modifier only seems to work on edges that are at the cube's actual edges. I tried assigning the loop edge to a vertex group, and then select "vertex group" in the bevel modifier, but to no avail. Please advise
Do 3 loop cuts (Ctrl+R) really close to eachother. Then select the one in the middle (Alt+Right Click) and scale it down (S).
Another option would be to separate the mesh by the selected loop. I don't remember if you can do that, but you can always select half the cube and make it a new object (P), then join it back again (Ctrl+J).
Excuse my poor answer but I don't have Blender on this computer to give you some pictures.