Fair enough, but I regard rhyming as essentially any repetition of sounds - whether they are at the end of words (moon and coin), the middle of words (moon and tooth), the start of words (moon and mire), or indeed at both the middle and end (moon and June). Clearly the last device is the most popular in songwriting, probably because it’s more appealing to vocalise, but it’s certainly not the only one used.
Stephen Fry takes you through all the different types of rhyme in his book ‘The Ode Less Travelled’, if my memory serves, but I accept it’s probably only of interest to those of us intrigued by songwriting (and poetry). Other texts on lyric writing are available, but are less easy reading in my experience. Before everybody else on here dies of boredom, here’s a final thought: even the complete repetition of a word is itself a rhyme, so the refrain line “Dirty old town, dirty old town” does actually contain a full rhyme. Some might even argue that’s three rhymes!