The tangent space is well-defined
Your proof is correct. Just a minor suggestion: You use the fact that if $\varphi : U \to V$ is a local parameterization around $x \in V$ and $\phi : U' \to V'$ is a local local parameterization around $x \in V'$ which is a restriction of $\varphi$, then $d\varphi_{\varphi^{-1}(x)} = d\phi_{\phi^{-1}(x)}$. This is quite obvious but should perhaps be mentioned.