Am I correct in assuming that any projective scheme $X \subset \mathbb P^n$ can be written as $X \cong \mathrm{Proj} (\Bbbk [x_0, \dotsc, x_n] / I)$ for some homogeneous ideal $I = (f_1, \dotsc, f_r)$ so that $X$ can be described concretely by considering the restriction of affine charts of $\mathbb P^n$ to $X$ and affine charts of $X$ can be given by $\mathrm{Spec} (A_i)$ where $$ A_i = \Bbbk [\tfrac{x_0}{x_i}, \dotsc, \hat{\tfrac{x_i}{x_i}}, \dotsc, \tfrac{x_n}{x_i}] / (f_{1, i}, \dotsc, f_{r,i}) $$ where $[x_0 : \dotsb : x_n]$ are homogeneous coordinates of $\mathbb P^n$ and $i$ runs from $0$ to $n$ and $f_{k,i}$ is the dehomogenization of the polynomial $f_k$ that appears as a relation in $I$?
When comparing schemes and varieties from the point of view of affine coordinate charts is the only difference that for schemes the $A_i$'s may contain nilpotent elements?