Why was the term "discrete" used in discrete logarithm? Is there anything especially "discrete" about a discrete logarithm? This is not a question of what is a discrete logarithm or why the discrete logarithm problem is an "intractable problem" given certain circumstances. I'm just trying to determine if there's some additional meaning to the term "discrete" as it's used in name discrete logarithm?

The definition of "discrete" is "individually separate and distinct". Could it be that the term "discrete" is a reference to the least non-negative residues of a modulus or the order of points for a particular cyclic group on an elliptic curve?
The word discrete is used as an antonym of 'continuous', that is, it is the normal logarithmic problem, just over a discrete group.

The standard logarithmic problem is over the infinite group $\mathbb{R}^*$, this group is called 'continuous', because for any element $x$, there are other elements that are arbitrarily close to it.

The discrete logarithmic problem is over a finite group (for example, $\mathbb{Z}_p^*$); in contrast to $\mathbb{R}^*$, we don't have group elements arbitrarily close together; we call this type of group 'discrete'.