Flexibility at least as it relates to "range of motion" is definitely context- and activity-specific. It relates to the nervous system's defense against stretching more than it relates to an actual quality of the tissue - your muscles have small sensors inside them that communicate with the central nervous system how much a tissue is being stretched, and your brain basically tells it to shut off stretching once you get to a certain point by contracting the muscle.
Stiffness, I guess, can be thought of in two ways. The way most people use it relates to a lack of flexibility, but stiffness as measured in the clinic by passive stretching may not be the same kind of loss of range of motion when someone is running. Activity can actually change your degree of defensiveness to stretch, so you might be able to achieve a range of motion while running that you could not while a doctor or therapist or researcher cranks on your hamstring. To me if a study is measuring flexibility as a function of range of motion, it probably has little relevance to runners.
Stiffness or elasticity as a science-y term is mostly about the degree to which something deforms when force is applied to it - this is PROBABLY more important for sprinters who deal with very high forces with each ground contact, but still matters for distance runners. Stiffer (more elastic) tissues are more energy-efficient, wasting less energy in tissue deformation and allowing that to return to the ground.
A good example of why flexibility is mostly mediated by the brain is when you do exams under anesthesia, where you try to take out conscious mediation of stretch by knocking out the patient - you can range folks without joint contractures MUCH further once the brain is zonked out :) Plenty of "inflexible" people become Gumby.