One definition of arithmetic circuits is as follows:
An arithmetic circuit $\Phi$ over the field $\mathbb F$ and the set of variables $X$ usually, $X = \{x_1, \dots , x_n\}$) is a directed acyclic graph as follows. The vertices of $\Phi$ are called gates. Every gate in $\Phi$ of in-degree $0$ is labeled by either a variable from $X$ or a field element from $\mathbb F$. Every other gate in $\Phi$ is labeled by either $\times$ or $+$ and has in-degree $2$.
A straight-line program seems to be a succession of basic operations without any if or while statement. So it seems to be a term from the associated free algebra expressed suitably "compressed" as an acyclic graph. But a (reference to a) nice definition of straight-line programs would already help me a bit with respect to this question.
What I really want to know is whether an arithmetic circuit is a special case of a straight-line program? If yes, then why is arithmetic circuit complexity considered so important, while even finding a reference for a proper definition of straight-line programs seems to be quite challenging? (I found a definition in The Extraordinary Power of Division in Straight Line Programs, but this is a paper from 2012, and the notion itself seems to be much older.)