So, there are two types of definitions of graded rings (I will consider only commutative rings) that I have seen:
1) A ring $R$ is called a graded ring if $R$ has a direct sum decomposition $R = \bigoplus_{n \in \mathbb{Z}} R_n$, where for all $m,n \in \mathbb{Z}, R_mR_n \subset R_{m+n}$.
2) A ring $R$ is called a graded ring if $R$ has a direct sum decomposition $R = \bigoplus_{n \in \mathbb{Z}} R_n$, where for all $m,n \in \mathbb{Z}, R_mR_n \subset R_{m+n}$, and $R_0$ is a subring of $R$, i.e., $1 \in R_0$.
In the second definition, is the additional condition that $R_0$ is a subring, i.e., basically the condition that $1 \in R_0$, redundant?
Yes it is redundant, you just need the definition 1). This is because:
if $ R_mR_n \subset R_{m+n}$ then $ R_0R_0 \subset R_{0}$ , thus $R_{0}$ is subring. Second we have $1=\sum x_n$ where there is only a finite number of non-zero $x_n$. Also note that $x_m = 1\cdot x_m=\sum x_nx_m$. By comparing degree we see that $x_m=x_0x_m$ and $x_0= 1 \cdot x_0=\sum x_n \cdot x_0=\sum x_n=1$, therefore $1 \in R_0$.