Frankly, I can never remember which type of verb Koichi’s categories refer to, and I’m fairly sure I learnt “group 1″ and “group 2″ the other way around.
Fortunately, I also learnt it as う-verbs and る-verbs, plus the two irregular verbs する and くる, which are unconfusable, since the defining characteristic is right in the name of the groups. I was never taught anything about ~いる or ~える.
る-verbs always end in る, the trick is in spotting which of the verbs ending in る are actually う-verbs. Like Kanjiman said, you get the feel for it after a while. By now, I can generally intuit which is which, but one trick I’ve developed (which may not be flawless) is to assume it’s a る-verb, then if the resulting conjugation sounds silly, it was probably a う-verb to start with. For example, はじまる -> はじまます sounds kinda silly. Reductio ad absurdum. =P
So yeah, あやまる is a う-verb. Whichever type that happens to be.