There is a very good reason for unsing $1$ when defining the characteristic: The advantages of using $1$ specifically is that
- it gives the largest characteristic among elements of the ring
- any other element is going to yield "characteristic" which is a divisor of the $1$-characteristic
- $1$ is the one element we can count on existing in the ring
For instance, in the ring $\Bbb Z/7\Bbb Z$, we get that $1+1+1+1+1+1+1 = 0$, but no other smaller sum of $1$'s is equal to $0$. Now, it also turns out that we get the same answer for any other non-zero number. For instance, $2+2+2+2+2+2+2 = 0$, but no other smaller sum of $2$'s is going to give $0$ (apart from the empty sum). In fields, you could use whatever non-zero number you want. This is where point 3 above comes in, though: In $\Bbb Z/7\Bbb Z$, using the number $5$ works, but in $\Bbb Z/5\Bbb Z$ it doesn't. Declaring that we will use $1$ means we aviod this issue.
As a different example, note that characteristic is also defined for non-field rigs (at least as long as they are commutative and unital). For instance, $\Bbb Z/6\Bbb Z$ has characteristic $6$, because $1+1+1+1+1+1 = 0$. However, if we use $2$ then we get $2+2+2 = 0$ and if we use $3$ we get $3+3 = 0$. So different numbers give different characteristics.
However, no matter which element you take, the "characteristic" that you get will be a divisor of $6$, and no matter which element you take, adding $6$ of them together will give you $0$, and no smaller number than $6$ works for all elements simultaneously. So the number $6$ still very much characterises the additive structure of the ring. So that's what we use.