That makes a lot of sense.Context does help. DracoSuave's thought experiment may be true in and of itself, but not within the context of Open Grave.
The first quote given by Ninja-to comes from a section titled The Nature of the Undead, which is divided into undead with souls and undead without souls. This section explains in brief that the soul is the consciousness of a living entity, but does not animate a living body and likewise does not animate the undead.
The second quote comes from a section titled Physiology. This section explains in more detail the relationship between body and soul. It explains that something called the animus governs the functions of the body, and that it acts as a conduit for the desires of the soul to influence those functions. (Might I suggest body is like hardware, animus like firmware, and soul like software and user?)
Think of the undead this way. With or without a soul, all undead are creatures that have had the animus restored. That force can be restored to no body at all (ghosts), to the same body (zombie or vampire), or to some other physical thing (helmed horror). Think of the undead as a disembodied or reembodied animus.
This explains how things that were never alive can become undead. A thing that was never alive can be classified as an undead construct if it is governed by the animus of a formerly living body. Hopefully, it also clarifies why magically animated constructs cannot be restored as undead, but physical things can become undead constructs when animated by the reembodied animus.
If a Warforged became a lich, what would it look like? All rusty?