Let us for a moment take the definition of “lighter than air” to mean “less mass per unit volume than air;” therefore by definition less dense. Then of course any object such as the ones under discussion here are not “lighter than air.”
The objects under discussion are of course mostly air. But the small proportion of their volume that’s not air (a solid such as metal) displaces that volume with something denser (namely, the solid).
Thus, any such structure must be denser than air.
Now I think you’re onto the definition that they’re using in the article: Namely, that the object is “lighter than air” meaning “less dense than air” so long as the object is not filled with air—that is, the object is in a vacuum.
What would happen if you released one of these objects from orbit? Might it gradually slow down due to aerobraking, but nevertheless come to float at some altitude above most of the atmosphere? I don’t think so, but it might take a very long time to come to earth.
The question hinges on what is meant by “volume”. In particular, does that volume include or exclude the large empty space between the metal parts? It’s crucially important because the dead volume is such an overwhelming majority of the total volume.
To answer your other question, no, it wouldn’t float because buoyancy needs the density of the surrounding in order to work. Up in space, the local air pressure is so low that it becomes much harder to be buoyant in it. Note that the object is full of holes, so whatever the external pressure is, will also be the internal pressure in the dead volume. The only difference is the (tiny) mass brought by the frame, which however will always add. In order to find some medium that this thing will be buoant in, you have to come up with a surrounding density that is greater than the metal itself (>>water, unless of course you don’t let the bubbles out). So the claim for “density” of this thing is somewhat misleading. At least in the case of styrofoam the dead space is sealed, so its mass/volume doesn’t change with air pressure or air composition so much.