Is this an equivalent definition for a graded ring?
Let $R$ be a ring. We say that $R$ is graded if there are subgroups $R_n, n\in \mathbb{Z}$ of $R$ such that given $x\in R$, there are $x_n \in R_n$ such that $x=\sum x_n$ where the sum is finite, i.e., there are only a finite number of non-zero $x_n$ such that this expression is unique.
The definition I am familiar is:
$R$ is graded if $R=\bigoplus R_n$ as abelian groups and $R_nR_m\subset R_{m+n}$
No, your definition is not equivalent to the usual one since it says nothing about the compatibility of the grading with the multiplicative structure of the ring.
For example let $k$ be a field and write $R=k[X]=\oplus R_n$ with $R_n=kX^{2n}\oplus kX^{2n+1}$.
Then $R$ is graded in your sense but not in the usual sense because $X^3\in R_1$ and yet $X^3\cdot X^3 \notin R_2$