"Differentially expressed" is just an arbitrary label we use to describe particular results. Most often, it refers to genes for which, in "some" comparison, a statistically significant change was detected. However, how the comparison is made, and between which groups (and even whether the genes are truly "significant" by some statistical parameter) will depend on the particular study.

I would say that in most literature, DEGs usually are genes with evidence of increase or decrease in expression in some condition as compared to other. (So, not "expressed in one condition as opposed to the other" as you propose). But again, DEGs really are what each paper defines them to be in their particular scenarios.