For $(U,x_1,...x_n),(V,y_1,...y_n)$, we may assume the charts are $(U,\phi),(V,\psi)$ respectively. Then $x_i:=r_i\circ\phi$, $y_i:=r_i\circ\psi$. In this sense, $x_i$ is not related to $y_1,...,y_n$ at all. However, in the https://people.maths.ox.ac.uk/hitchin/hitchinnotes/Differentiable_manifolds/Chapter_3.pdf p.60 the last but one line, it uses $y_i(x_1,...x_n)$. I don't understand why we can see $y_i$ as a function with variables $x_1,...x_n$.
Moreover, on some textbooks, a differential $\omega$ on $U\cap V$, has local expression $fdx_1\wedge...\wedge dx_n$ associated with $(U,x_1,...x_n)$ and has local expression $gdy_1\wedge...\wedge dy_n$ associated with $(V,y_1,...y_n)$, then they claim $\omega=fdx_1\wedge...\wedge dx_n=gdy_1\wedge...\wedge dy_n$ on $U\cap V$. However, they are in different charts, it doesn't make sense to say they are equal, right?