Say we have two affine plane curves that pass through $P=(0,0)$, namely $X=Z(f), Y=Z(g)$. In the begging of the book "Intersection theory" by Fulton he defines the intersection multiplicity of $X,Y$ at $P$ as: $dim_k\mathcal O_P$ where $\mathcal O$ is the structure sheaf of the (scheme-theoretic) intersection $X\cap Y$.
However, In chapter I, section 5 of Hartshorn's "Algebraic Geometry" he defines the the intersection multiplicity as $length \mathcal O_P$ over $k[x,y]_P$, the localized coordinate ring of $\mathbb A^2$.
Why are these definitions equivalent? When we define more general intersection multiplicities, such as when $M$ is a good graded module over a good graded ring $S$, then also Hartshorne defines the multiplicity of $M$ at some minimal prime of $Ann(M)$ as the length of $M_P$ over $S_P$. Can this definition be reformulated in terms of dimension? What are the essential differences in these definitions?