Torsors in Algebraic Geometry?

I think I am confused about some terminology in algebraic geometry, specifically the meaning of the term "torsor". Suppose that I fix a scheme S. I want to work with torsors over S. Let $mu$ be a sheaf of abelian groups over S. Then my understanding is that a $mu$-torsor, what ever that is, should be classified by the cohomology gorup $H^1(X; mu) cong check H^1(X; mu)$.

Now suppose that $mu$ is representable in the category of schemes over S, i.e. there is a group object $$mathbb{G} to S$$
in the category of schemes over $S$, and maps (over S) to $mathbb{G}$ is the same as $mu$. Lots of interesting example arise this way.

I also thought that in this case a torsor over S can be defined as a scheme $P to S$ over S with an action of the group $mathbb{G}$ such that locally in S it is trivial. I.e. there exists a cover $U to S$ such that
$$ P times_S U cong mathbb{G} times_S U $$
as spaces over S with a $mathbb{G}$-action (or rather as spaces over U with a $mathbb{G} times_S U$-action).

The part that confuses me is that these two notions don't seem to agree. Here is an example that I think shows the difference. Let $S= mathbb{A}^1$ be the affine line (over a field k) and let $x_1$ and $x_2$ by two distinct points in $S$. Consider the subscheme $Y = x_1 cup x_2$, and let $C_Y$ be the complement of Y in S. Let $A$ be your favorite finite abelian group which we consider as a constant sheaf over S. Then we have an exact sequence of sheaves over S,
$$0 to A_{C_Y} to A to i_*A to 0$$
Where $i_*A(U) = A(U cap Y)$. I believe the first two are representable by schemes over S, namely $$C_Y times A cup S times 0$$
and $S times A$, where we are viewing the finite set $A$ as a scheme over $k$ (and these products are scheme-theoretic products of schemes over $spec ; k$).

In any event, the long exact sequence in sheaf cohomology shows that
$$H^1(S; A_{C_Y}) cong check H^1(S; A_{C_Y}) cong A$$
and it is easy to build an explicit C$check{text{e}}$ch cocycle using the covering given by the two opens consisting of the subschemes $U_i = S setminus x_i$, for $i = 1,2$.

Now the problem comes when I try to glue these together to get a representable object over S, i.e. a torsor in the second sense. Then I am looking at the coequalizer of
$$C_Y times A rightrightarrows (C_Ycup C_Y) times A$$
where the first map is the inclusion and the second is the usual inclusion together with addition by a given fixed element $a in A$. This seems to just gives back the trivial "torsor" $C_Y times A$.

Am I doing this calculation wrong, or is there really a difference between these two notions of torsor?