Let $k$ be a commutative ring. Let $M=\oplus_iM_i$ be a $\mathbb Z$-graded $k$-module. There is the symmetry algebra of $M$, defined as
$$\mathrm{Sym}^{\bullet}(M):=T^{\bullet}(M)/\langle x\otimes y-y\otimes x~|~x, y\in M\rangle$$ and there is the so-called graded symmetric algebra of $M$, defined as
$$\mathrm{Sym}^{\bullet}(M):=T^{\bullet}(M)/\langle x\otimes y-(-1)^{|x||y|}y\otimes x~|~x, y\in M\rangle,$$
where $|x|, |y|$ are the degrees of $x$ and $y$, respectively. Both are graded algebras with grading defined the same way- the grading is inherited from the tensor algebra.
Why is the latter called graded symmetric algebra, when the first one is graded, too?