It has to do with which syllable is stressed. The last stressed syllable in the word (or in the line of poetry), and everything after it, has to be the same. So the "-eacher" in "teacher" and "reacher" are identical, and it's a good rhyme. To rhyme with "hex", the word has to end in "-ex." You can fudge the consonants after the stressed vowel just a bit, and get something that's not quite a rhyme, but good enough for, say, a HN comment. But you can't fudge the stressed vowel at all.
What I was describing (where the last stressed vowel matches but the rest doesn't all match) is called assonance. This isn't even assonance, much less a rhyme. If you have a source describing this kind of not-a-rhyme from any period of English versification, I'd be interested to see it.
But in your examples, the last stressed vowel and the unstressed one, both match: reacher, teacher. Therefore I don't follow your comment.
There is a kind of assonance (or whatever) in English where the stressed one can differ. The juxtaposition of such words or endings can sound good. For instance "pitter-patter". This can be exploited to create a near rhyme, like my teacher and catcher verse.
Moreover, if it is the unstressed one that doesn't match, that is no longer the case. We cannot rhyme "teacher" with "teapot" as line endings.
states the same conditions you have given for perfect rhyme: stressed vowel and following unstressed, if any, are a match. Then it goes into the exceptions.
