Let's take two people we would call 'teachers' in a literal and straightforward sense:
(1) Master Kong, who is professionally a teacher. Indeed, we can say he is preeminently a teacher; Confucius is, we might say, a teacher among teachers. He is able to teach, and, what is more, teach how to teach, by example and word, and he is able to teach an immense amount with very little. He is very, very good. He is so astoundingly good that he started a teaching movement nearly twenty-five hundred years ago that has never completely died and continues to inspire.
(2) So as not to hurt anyone's feelings, especially my own, we will make up another professional teacher, Joe Bore. He is as boring as his name; he is lackluster and mediocre and uninspiring and pedantic without being especially well informed. He is unoriginal, unwitty, unrelenting in his tiresome, endlessly tiresome, droning. His lectures are confusing. His assignments are activity without point. When students ask questions, the answers make things more difficult to understand. No, this is not a self-portrait, although there are days when it feels like one. But, in any case, Joe Bore teaches and continues to teach and students usually manage to learn something of some sort.
Both Master Kong and Joe Bore are teachers, undeniably. They teach, and that is their role. In neither case is the term being used figuratively, and in neither case are we really stretching the term or playing on ambiguities. And yet that does not seem to be all there is to say, does it? To say that Master Kong is a teacher and that Joe Bore is, too, is entirely right, and yet seems in some sense to understate the extent to which Master Kong is more properly a teacher than Joe Bore. Joe Bore may be a teacher, but Master Kong is a teacher. Confucius is not just a teacher, he is a teacher, emphasis needed.
We often distinguish the intension and the extension of a term. If we ask what is meant by a term, we can always give the content of it; this is the intension -- 'teacher' means one who guides another in study so that they might understand. But we can also answer by indicating what is covered by the term, the extension: Confucius and Joe Bore and all the others on our list are capable of being called 'teacher'. What is less often considered is the relationship between intension and extension, and as we see in the case of a term like 'teacher', while meaning of the term (intension) defines a list (extension) that includes both Confucius and Joe Bore, the intension is more fully realized in Confucius than Joe Bore -- he's a better example of a teacher, he is a more central case, he fills the role laid out by the intension of 'teacher' much better than Joe Bore does. Even though they are both properly called 'teacher', Confucius is more properly called 'teacher'. An intension does not need to 'fit' everything in its extension equally well; but the differences of degree of fit is something we can pick up on and express. It is part of the meaning.
I deliberately picked 'teacher' because it is not the sort of term one might immediately recognize to have different degrees of fit between content and what is covered by it, but which nonetheless has clear examples. Most substantive terms whose intensions involve some kind of role will work the same way. But there are lots of others that are even more obvious cases. For instance, color adjectives: we are sorting color chips into the boxes 'red' and 'not red', and we put two color chips into the red box -- one dusky, definitely red, but getting a bit gray, and one brilliant, richly saturated with red hue. They are both definitely red, but the latter is a stricter fit to what is meant by 'red'. And, of course, as happens with colors, so things go with all adjectives capable of a gradation analogous to saturation of hue -- the intension grades over the extension, so to speak.