In the literature on algebraic curves, I have found two definitions of places on a curve.
Let $K$ be an algebraically closed field.
Let $\mathcal{C}$ be an irreducible curve defined by $f(x,y) \in K[x,y]$. Let $F(x,y,z) \in K[x,y,z]$ be the corresponding homogeneous polynomial.
Definition 1:
An affine local parametrization (ALP) of a curve is a pair $(A,B) \in K((x))^2$ such that $f(A,B)=F(A,B,1)=0$ and not both $A$ and $B$ are in $K$. We say that two ALPs $(A,B)$ and $(A',B')$ are equivalent if there exists $C \in K((x))$ such that $(A',B') = (A(C),B(C))$. Every ALP is equivalent to an ALP of the form $(t^n, a_1 t^{n_1} + a_2 t^{n_2} + a_3 t^{n_3} + \cdots)$, which we can use as representative. If the integers $0 < n$ and $0 < n_1 < n_2 < \cdots$ have no common divisors then the ALP is said to be irreducible. Then a place of $\mathcal{C}$ is an equivalence class of irreducible ALPs.
found in e.g. Robert Walker's Algebraic Curves
Definition 2:
Consider the algebraic function field $R=K(x)[y]/f$. A place of $R$ is a valuation ring $P \subseteq R$, i.e. a subring such that $K \subseteq P \neq R$ and for all $u \in R$, either $u \in P$ or $u^{-1} \in P$.
found in e.g. Serge Lang's Introduction to Algebraic and Abelian Functions
I am sure the two definitions must be related in some way, given that $K[[x]]$ intervening in Definition 1 is a valuation ring. My question is: are these two definitions the same? If so, how can one see that?