Intuition behind straight-line programs

Question

Wikipedia defines straight-line programs in the following manner:

In mathematics, more specifically in computational algebra[disambiguation needed], a straight-line program (SLP) for a finite group G = 〈S〉 is a finite sequence L of elements of G such that every element of L that belongs to S, is the inverse of a preceding element, or the product of two preceding elements. An SLP L is said to compute a group element g ∈ G if g ∈ L, where g is encoded by a word in S and its inverses.

This is a fairly confusing definition to unpack. However, in some meaningful sense, I believe the main idea here is that there are no comparisons allowed. Since there are no comparisons, the "computation tree" ends up being a straight line - hence the name "straight-line program."

In modern terms, is this the correct idea? Or are there even more subtle restrictions than this on what sorts of programs are allowed to be straight-line programs?