Just because a wood springs back without deformation after being bent doesn't necessarily mean it is elastic. There are species, Black Cherry for instance, that will spring back quite well after being bent but take it too far and presto, chrysals all over the place. You can also find low in elasticity wood in species known for their elasticity. Wood high in elasticity can be bent pretty severely and they will take set but will not chrysal, you can see this in low density Yew. Elastic failure in compression is essentially the wood cells crushing.
I am quite sure that elasticity also plays a part in tension as well, just not as much as in compression.