How could a teacher clarify to the students the difference between the " discursive" and " argumentative" essay?

Most essays serve a single purpose which is expressing a personal opinion on an issue. All these different labels for the essays do not change this fact. It's how you present your argument that gives your essay its character. If you use description so it's descriptive. If you use logical argument, so it's argumentative. All in all, writing in general is to voice out a personal stand or idea but in the form of scripture, but in academia it's more disciplined as to take the form of an academic essay.