The result does not depend on the disc $D$ that you choose. You can see this $n$-th twist as a self-diffeomorphism $\varphi$ of the solid torus $S^3 \setminus K$. If you pick two distinct discs $D, D'$, these are isotopic in this solid torus and hence the two self-diffeomorphisms $\varphi, \varphi'$ that they produce are isotopic. Therefore there is an isotopy relating the two links $\varphi(L\setminus K)$ and $\varphi'(L\setminus K)$ in the solid torus $S^3 \setminus K$, which extend to an isotopy of the whole twisted links in $S^3$.

The result does not depend on the diagram, because the twisting operation does not depend on the disc $D$ with $\partial D = K$ that you choose. You can see this $n$-th twist as a self-diffeomorphism $\varphi$ of the solid torus $S^3 \setminus K$. If you pick two distinct discs $D, D'$, these are isotopic in this solid torus and hence the two self-diffeomorphisms $\varphi, \varphi'$ that they produce are isotopic. Therefore there is an isotopy relating the two links $\varphi(L\setminus K)$ and $\varphi'(L\setminus K)$ in the solid torus $S^3 \setminus K$, which extend to an isotopy of the whole twisted links in $S^3$.