For a quad based topology, you would divide the cube to 3 face loops, so you can lower the middle one. Then you would divide it also vertically, to at least 2 face loops*, so you can lower it to the nearest loop, using vertex snapping and ⎇ AltE Extrude Manifold:

*-I used 3 loops to get squares, but it's not very important, especially if you're about to scale the loops later in order to get the specific dimensions you want:

But it seems you want to get a minimum topology, meaning as few polygons as possible, and in this case you can just create a single face, and then extrude it:

X key to access the delete/dissolve menu, switch between vert, edge, face selection mode using 123 keys. Once you have the cube you can just dissolve unnecessary edges and vertices, as you saw at the beginning of the last gif.