You are right: Eisenbud's definition is not the standard one .
Standard definition : a Cartier divisor on the scheme $X$ is a global section $D \in \Gamma(X,\mathcal K^*_X/ O^*_X)$, where $\mathcal K$ is the sheaf of rational functions on $X$.
In the affine case $X=Spec(R)$, this translates into $D\in (TotR)^*/R^*$, where $Tot R$ is the ring of fractions $S^{-1}R$ with $S=$ the set of non zero-divisors.
In practice, $D$ is given by an open covering$(U_i)$ of $X$ and rational functions $s_i\in \mathcal K^* (U_i)$, such that $s_i=g_{ij}\cdot s_j$ on $U_i\cap U_j$ and $g_{ij}\in \mathcal O^*(U_{ij})$.
To these data you associate Eisenbud's invertible ideal sheaf $\mathcal O(D)\subset \mathcal K_X$, characterized by: $\mathcal O(D)|U_i=\frac{1}{s_i}\mathcal K_X|U_i$.
This correspondence between Cartier divisors and invertible subsheaves of the rational functions goes over to isomorphism classes and yields an isomorphism between $Cacl(X)$, the additive group of classes of Cartier divisors, and $Inv(X)$, the multiplicative group of classes of invertible fractional ideal sheaves, in other words between the standard point of view and Eisenbud's.
I think Eisenbud adopted his point of view because in the affine case $X=Spec(R) $, fractional ideals of $R$ are a fairly elementary concept and are already known to students having followed a first course on algebraic number theory.