From Hartshorne I.6:
If $v$ is a valuation, then the set $R = \{x \in K | v(x) \geq 0\} \cup \{0\}$ is a subring of $K$, which we call the valuation ring of $v$... A valuation ring is an integral domain which is the valuation ring of some valuation of its quotient field.
What I am confused about is when someone makes a statement like:
If $R$ is a Noetherian regular local domain of dimension 1, then $R$ is a discrete valuation ring with valuation $v$
Couldn't there be many possible valuations $v$ which give rise to $R$?
For example, if one is talking about a projective variety $X$ and a hypersurface $Y$ with generic point $\eta$ and uses the above fact on $\mathcal{O}_{X,\eta}$ to obtain the valuation $v_Y$ associated to the order of vanishing of a rational function along $Y$, how do we know that there aren't any other valuations taking different values that give back the same ring?
We can take $v$ as $v(x)=n$ iff $x\in {\frak m}^n\backslash {\frak m}^{n+1}$ for local ring $(R,{\frak m})$.
And in your example, $X$ is 1-dimensional so $Y$ is just one point.