I apologize for writing something wrong in my comment. Here is a quick amplification of the valid part of my comment. Let $X$ be a scheme that is finitely presented over $\text{Spec}\ k$ for a field $k$. Then Grothendieck's definition of $\text{CH}^1(X)$ as the first graded piece of the gamma filtration is equal to $\text{Pic}(X)$. This is explained, for instance, in Manin's "Lectures on the K-functor in algebraic geometry." However, Fulton's definition is insensitive to nonreduced structure on $X$ and to seminormalization. Thus, Fulton's definition equals $\text{CH}^1$ of the seminormalization of the reduced scheme of $X$.

However, as noted by others, the two definitions do appear to agree for the line with doubled origin. The Picard group is a free cyclic group. I do not know if the two definitions always agree for smooth, finitely presented, but possibly non-separated $k$-schemes. You might check SGA 6, since the seminar participants worked there in great generality (and later authors, such as Thomason, worked in even greater generality).