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).