Let us not perpetuate a misconception of what is stiffness. Indeed, the modulus of elasticity (Young's modulus), as mentioned in your citation, is important to stiffness. But stiffness of beams (structural elements that bend) it is also dependent on the moment of inertia (second moment of area) which for a rectangular cross-section varies as the third power of thickness. (If the thickness doubles, the stiffness is multiplied by eight.) See this article.
Of course, a thick board is stiffer than a thin board. Using "stiffness" is an error by the citation.
The answer to your question
By flexibility I mean e.g. how much will a shelf of a given thickness and grain orientation bend
is actually: modulus of elasticity.
For more information on the bending characteristics of wood, you could look at this article. It shows how wood beams typically act under loads and eventually break.
Further research on the net reveals that the MOE grows larger as the moisture content lessens (moist beams sag more than dry beams). MOE can actually increase by 50% between green wood and wood dried to 12% moisture content.
The MOE also varies greatly from species to species and greatly affects the amount of deflection in a beam or shelf. See this to see that MOE for western red cedar is about 2/3 of that for red oak and only 1/2 of that for shag bark hickory. A book shelf made of the cedar will sag 50% more than one of the same dimensions made from oak.
And then there's the issue of wood continuing to sag with no increase in load over a long period of time. We have all seen book shelves that continue to sag even when all of the books are removed and there is no load on the shelves. This can happen even if the shelf was not loaded beyond its elastic limit. Wood behaves differently from metals, which, incidentally, behave differently from one another.