Can someone explain the two definitions in relation to each other?
Is a claw-free permutation a permutation without a trapdoor?
For your convenience, here's the definition of a "claw-free permutation":
Let $f_1,f_2$ be permutations. A triple $(x,y,z)$ is called a claw iff $f_1(x)=f_2(x)=z$. A pair $f_1,f_2$ of permutations is called claw-free iff there exists no efficient algorithm that can find a claw for these permutations.
The definition of a trapdoor permutation is a bit lengthier and thus only given as a reference.