زَيْدٌ الْمُعَلِّمُ — Zayd is the teacher. Both terms are definite.
In one context, Zayd may be the mubtadaʾ and «الْمُعَلِّمُ» the khabar; in another, the second expression might be parsed as badal or ʿaṭf bayān.
The textbook’s permission to select either term is a rule of basic nominal predication, not permission to ignore discourse meaning or competing analyses.