In the first one, it's general/generic. An elephant is the largest land mammal. A teacher's role is... There would be no change in meaning if you changed it to "The."
In the second one, I would not have written it that way. I'd say either "her role as teacher" or even "her teaching approach."
In truth, I don't find "teacher role" to be very natural and I couldn't think of a natural sentence for it when I saw the first post. But there have been so many times when I've said "I can't think of a time when this would work" only to have perfectly good and natural examples given as a rebuttal. Hence, my request for context.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.