I can never remember which of Koichi’s groups is which, but I learnt it as う-verbs, る-verbs and irregular (which consists only of する and 来る). I’m sure you can match those to Koichi’s types well enough. Japanese for Busy People uses Regular I (= う-verbs), Regular II (= る-verbs) and Irregular, but personally I prefer group names that are actually descriptive of the verbs in the groups.

Japanese names are 一段 (いちだん = る-verbs) and 五段 (ごだん = う-verbs). Don’t know the etymology behind that, but the way I remember it (which could be correct) is that there’s one way to form the plain past tense with 一段 verbs (stem+た), while there’s five ways to form it with 五段 (～った、～んだ、～いた、～いだ and ～した). Can’t tell if there’s an actual group name for the irregulars, but hey, there’s only two of them.

yes, る and う verbs are the way to go, IMO!  much easier!