I see two definitions of a regular semiring.
$1^{st}$ definition: A semiring $R$ is called a regular semiring if for every $x\in R$; $x = xax$ for some $a\in R$.
$2$nd definition: A semiring $(R, +, .)$ is called a completely regular semiring if every element $x$ of $R$ is completely regular. An element $x\in R$ is called a regular if there exists an element $a\in R$ such that $x= x+a+x$, $x + a = a + x$ and $a$ is called completely regular if it additionally satisfies that $x(x + a) = x + a$.
Are these two definitions of the regularity of a semiring equivalent? Also, how to define a completely regular semiring in sense of the first definition? Edited A reference for the second definition is here
The two definitions of regular are obviously not equivalent.
By the second definition, every ring is a completely regular semiring (just let $a=-x$), but not every ring is regular in the first sense (which is also called von Neumann regularity.)
The second definition of regularity looks like it's talking about regularity of the $+$ structure, while the first is regularity of the $\cdot$ structure.
I have no clue about the condition $x(x+a)=x+a$. It is new to me. It would help if you actually mentioned your sources. In fact, that would be a good habit to start practicing right now...