Is the difference simply that, general v. specific?
No, but it is the starting point for teaching English as a foreign language.
I suggest starting out with only general vs. specific usage, and then adding other rules bit by bit (giving precedence to those most used):
- general vs. specific
- always before ordinal numbers (but never with possessives)
- always before superlatives (but never with possessives)
- always before river names
The point is to avoid drowning the learner in rules, thus only giving more rules when the previous are understood and assimilated. I recommend getting a list of rules from a good grammar book and choosing the ones more relevant for the learner's level.
Omitting article “the” in front of plural nouns
It seems I missed the 'question' in the title and answered only the last one.
'The' isn't used with plurals when that plural implies a general reference, only when that plural implies a specific group.
Students learn better when teachers are passionate about teaching.
is the equivalent to: "Any student learns better when any teacher is passionate about teaching."
The students learn better when the teachers are passionate about teaching.
implies that you're not talking about students in general but a particular group, for example those who study in a certain school or at a certain level; and the same goes for the teachers: you don't mean teachers in general.
Without a context that clarifies whether the sentence relates to a specific group of students/teachers, the first sentence (zero article) is preferred.