Why was the term "discrete" used in discrete logarithm?
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'.