Category of $\textbf{Ring}$. I learned that $\textbf{Ring}$ is the category of all nonzero rings with $1$, where morphisms are ring homomorphisms that send $1$ to $1$, in the "Abstract algebra, Dummit and Foote". But I think that it is more reasonable to define it as the category of all rings, where morphisms are ring homomorphisms. Why is $\textbf{Ring}$ defined as above?
 A: Both definitions are plausible and have their respective applications. It is only a matter of context.
There is a category of rings with unity as well as a category of rings not necessarily with a multiplicative unity. In the category of rings with unity it makes sense that the morphisms in that category should preserve that additional structure (the $1$), therefore sending unit on unit.
Notes:
That these categories really are different can be seen already from simple examples. Consider for example a ring homomorphism (without unity) $f:\mathbb{Z}\rightarrow\mathbb{Z}$. Then $f(z)=k\cdot z$ for some $k\in\mathbb{Z}$, where $k=f(1)$.
But in the category $\mathrm{Ring}$ the only morphism $\mathbb{Z}\rightarrow\mathbb{Z}$ is the identity, since $f(1)=1$. Also any other morphism $\mathbb{Z}\rightarrow R$ where $R$ is a ring with unity is uniquely determined: $\mathbb{Z}$ is an (the) initial object in $\mathrm{Ring}$. It is not in the category of rings without unity.
E.g. in fields such as representation theory it is mostly desirable for rings to have a unity, since there one is interested in $R$-modules. 
The identity element in the ground ring induces the identity map of a module $x\mapsto 1\cdot x$.
