My structural engineering is real rusty, but I believe the both cross sections will have the same compressive strength (psi), but the taller cross section will have a higher higher moment strength (or flexural strength may be a more correct term).
Compressive:
Flexural:
The compressive strength of a material is based on straight force (pounds) over a given area (square inches, for example). So even with equal compressive strengths, the wide, short board offers a larger surface, and thus can support more pounds. More square inches, more pounds.
The moment strength is resistance to a combination of compressive and tensile forces (pushing and pulling), depending where in the board you're looking. It ths sketch above, the top part of the board is under compression and the lower part is under tension. I believe strength is gained by going to a taller cross section faster than it is by using a wider one. Like I said, I'm real rusty on this stuff, so I could stand corrected.
In other words, I believe the shorter board (the 6 x 2) would have more strength in the first picture, but the taller board (the 12 x 1) would be stronger in the second example. Of course, this doesn't account for racking and other sideways forces.