Say $V$ is an $n+1$-dimensional vector space with basis $\{e_1,\dots,e_{n+1}\}$. Then we can take the projective space of this $\Bbb P(V)\cong \Bbb P^{n}$.
Consider the affine open $U_0$ where $z_0\ne 0$. It seems I can write the homogeneous polynomials that gives me the following point of $U_0$, say with homogeneous indeterminates coming from $K[X_0,\dots,X_n]$: $$[1:z_1:\dots:z_n]\longleftrightarrow V_h(z_1X_0-X_1,\dots,z_nX_0-X_n).$$
If someone tells me the homogeneous coordinate relative to $e_i$ is nonvanishing, what does that canonically refer to? I could guess that they want me to actually take that identification $\Bbb P(V)\cong \Bbb P^n$, and then $e_i$ corresponds to the homogeneous coordinate in the $i^{th}$ position, but I am not sure this is the canonical meaning (if there is one in the literature).
What does it mean to say the homogeneous coordinate corresponding to $e_i$ when we take $\Bbb P(V)$ for $V$ having basis $\{e_1,\dots,e_{n+1}\}$?
The homogeneous coordinate corresponding to $e_i$ means $z_{i-1}$, if your point $z$ is $[z_0:z_1:\dots:z_n]$. (It's $z_{i-1}$ instead of $z_i$ because you've made the unfortunate choice to index your basis starting at $1$ but your coordinates starting at $0$.)
Note that this coordinate $z_{i-1}$ is not well-defined, since you could multiply all the coordinates of $z$ by a nonzero scalar $\lambda$ to get the same point in $\mathbb{P}^n$. However, it is well-defined whether it vanishes.