Let $L$ be a line bundle on a smooth curve $C$. If $L$ is rank $r+1$, we get a induced map $C \rightarrow \mathbb P^r$. Why do many sources (as Arbarello, Cornalba, Griffiths, Harris), define this map as
$$\begin{aligned}\phi :& C \rightarrow \mathbb P|L|^*\\&p \mapsto \{s\in |L|:s(p) =0\}\end{aligned}$$
instead of
$$\begin{aligned}\phi :& C \rightarrow \mathbb P|L|\\&p \mapsto (s_0(p),...,s_{r+1}(p))\end{aligned}$$
where $(s_0,...,s_{r+1})$ is just any basis of $|L|$? Is it just to avoid picking a basis?
Question 2) If we compose the second map with an isomorphism of $\mathbb P|L|$ to $\mathbb P|L|^*$ given by picking the same basis $(s_0,...,s_{r+1})$, does that equal the first map?
$\phi$ should always be viewed as a map to $\mathbb{P}|L|^*$, because every non-base point $p\in C$ natually gives a codimension $1$ hyperplane in $|L|$, i.e. $\{s\in|L|: s(p)=0\}$.
This holds not only for $C$ but also for any variety over an algebraically closed field $k$ and any closed point $p$ which is not a base point of $|L|$.