Mostly ignoring the question of whether I'd climb this tree or not (Personally I wouldn't. No way! Seeing that one picture of the defect, I just simply wouldn't feel safe on an emotional/intuitive level), I'll instead make some comments that might put the question of whether 10% wall thickness is "safe" or not into a different light. As Ekka noted, the TCIA reference mixes concepts here; diameter on the one hand, and wall thickness on the other. If you think about it, wall thickness really relates to radius of the tree, and so to be able to be compared to diameter on an apples to apples basis, this value must be doubled.
So this "10% wall thickness" is really 20% of diameter, and if you do the arithmetic of calculating the percentage of good wood associated with this remaining ring, it's 36% of the original total.
So what TCIA appears to really be saying is that with 36% of the wood remaining, you still have (nominally) 50% of the "strength" of the tree. (By the way, it's not completely clear exactly what they mean by "strength"; ability to withstand a bending moment, or ability to withstand a compressive load, or ...?)
Bypassing the question of what they mean by "strength", certainly it's more intuitively believable that 36% of the remaining wood could yield 50% strength than it is to say that "10% wall thickness" does so. Then, when you consider that in a bending scenario, the fibers of the outside of the tree are under the greatest stress (the ones on the side in the direction of the bend are in compression, and the ones on the opposite side are in tension), and these stresses decrease as you move to the center of the tree (the fiber at the geometric center is under neither tension nor compression), having the 36% of the tree being those fibers located in the areas where the stress is greatest renders this even more intuitively believable. That is, the fibers where the wood is good are in exactly the right location to do the most good.
So from an engineering mechanics and strength of materials viewpoint, I think this 10% wall thickness is probably a pretty good number. This assumes, that the tree is a nice uniform hollow cylinder, without major imperfections. All bets are off if the tree has major defects that compromise this critical load-handling capability of the outer fibers.
By the way, adkranger's point about hollow versus solid is valid. As stated above, the distribution of loads varies from maximum at the edges to zero in the center, so the presence of additional fibers beyond the 10% thick wall certainly helps, albeit to a decreasing extent as you move to the center.