In my algebraic geometry class, the dimension of an affine variety $X=V(I)$ was defined as the supremum of the length of chains of prime ideals in the coordinate ring $R=k[x_1,\ldots,x_n]/\sqrt{I}$, or, the Krull dimension of $R$. A few minutes later, we were given the definition in terms of transcendence degree: the dimension of $X$, if $X$ is integral, is the transcendence degree of $K(R)$ over $k$. I have also seen the dimension defined in terms of the degree of Hilbert polynomials (Theorem C, page 225 in Eisenbud).
My question: which of these came first and why? Why were the other definitions developed? What did they allow us to do more easily or effectively that previous definitions did not?