Is $\{\varepsilon\}$ a conventional way to mark the empty language?
No, that's the language containing one string: the empty string, which is very much distinct from the empty language. The finite state machine on the left accepts language $\{\}$, the finite state machine on the right accepts language $\{\varepsilon\}$. They are not the same.

If a lot of students are confused maybe explicitly show the difference in an exercise/during a lecture.

As an example, we could use the language as a password check. Anyone submitting an accepted word would be let through. Then $\{\}$ means &quot;let no one in&quot; and $\{\varepsilon\}$ means &quot;let anyone that submits a blank password in&quot;. Those are quite different indeed!