Reading the paper, plastic buckling in this context means the collapse of micro-scale cells forming the structure of the wood, causing local buckling of the cell walls. The net effect of this is a global failure which presumably looks like a crushed band in the sample. In a sense, each individual cell wall is undergoing a constrained version of elastic buckling. The overall sample permanently deforms as a result, hence plastic buckling.
The overall effect of plastic buckling is in contrast to elastic buckling, which is a global bending phenomenon caused by an eccentric loading instability in sufficiently slender columns and plates loaded in compression along their long axes. If the centers of pressure of the two sample-fixture interfaces don't align perfectly along the compression direction, there is a slight eccentricity of the loading. The eccentricity leads to the formation of a moment couple, subsequent bending, and ultimately elastic buckling. Note that there is no way such perfect alignment can occur, and so elastic buckling should always be considered in slender objects loaded in compression along their length.
As for the significance of the graph, it appears to be showing that density and yield strength of balsa are linear over the given range of densities, regardless of other common considerations or test methods. Note that the material can undergo special failure modes not available to monolithic materials due to its porous microstructure. Balsa has a hollow cell microstructure, and individual cells can collapse in one of two ways. The first is by plastic buckling, where cell walls buckle, causing axial cell collapse. The second is by kink-band formation, or a coordinated cell collapse where individual cells shear, appearing to tip over. Because these failure mechanisms don't require much physical space to occur around the sample, unlike with elastic buckling, they allow balsa to fail regardless of confinement. As a result, constraint has no effect on the density-yield relationship. To support this, from the conclusion section of the paper,
Independence of strength from confinement is attributed to the highly
porous cellular microstructure of balsa.
The linearity of the relationship is explained by the fact that different mechanisms come into play at different densities. Despite that they may individually have nonlinear density-yield relationships, as in the first equation you posted, at some density another mechanism takes over and becomes energetically favorable. On either side of that density, different mechanisms will be observed at failure. It appears that, over the range of samples observed, the mechanisms coincidentally line up. At different densities, different failure mechanisms are seen because they are more energetically favorable. To support this, again from the conclusions section,
The post-mortem SEM examinations reveal that failure mode transition
from elastic/plastic buckling to kink band formation occurs as the
To sum up, the samples failing by plastic buckling aren't buckling like a beam does, but rather there is a coordinated buckling of a microstructural feature resulting in permanent deformation of the sample. Because balsa has a hollow cell microstructure, some failure modes can occur even when the sample is mechanically confined. Also, different failure modes occur at different densities in such a way that density happens to form an overall linear relationship with yield-strength.