Similar to how an NaCl-structure can be described as just a primitive cubic structure if all the lattice points were the same element or ion. I'm asking if there any such alternative for the CsCl-structure.
You certainly can construct what is called a primitive cell containing one atom for a bcc lattice. There could be more than one way, but here is my method.
Start with the usual cubic representation. Pick any face and connect the midpoints of its edges to make a smaller square. Do the same with the opposite face. This gives a pair of squares which are then the bases of an elongated square prism, whose lateral edges connect the two opposing faces you "contracted". This cell, which contains just the central atom of the original bcc structure, is a primitive cell. In this cell the lateral edges, which we did not "contract", are about 1.41 (actually, the square root of 2) times as long as the basal edges.
How does this replicate to make the lattice? If you just scoot the primitive cell over to "catch" a neighboring atom on either side of the lateral faces, you find that you must move the adjacent cell "up" or "down", in the direction perpendicular to the bases, to get the second atom centered properly. The "up"/"down" move is half the height of the prism. But once you see this, you can repeat this process and get all the atoms involved in the entire cubic representation.
Last question: I said the primitive cell is an elongated square prism. It no longer has cubic symmetry. Where did the cubic symmetry go? The answer is you can't make a primitive cell in the bcc lattice with the full cubic symmetry. Instead, note that when you modified the cube you had the choice of three pairs of opposite faces to "contract". Thus there is really a combination of three primitive cells with different orientations. The cubic symmetry emerges from the combination rather than any one primitive cell.
Yes there is. In your particular case, it is also the primitive cubic lattice.
A hint is in the name - BCC = Body Centered Cubic Any arrangement that is BCC is also primitive cubic, for the isolated constituent building blocks.