The rules you stated aren't rules. They never were rules and they don't make any sense as rules. Other and another are semantically exclusive of each other.
Another is concatenative. It adds the new thing to a set of existing things that are receiving focus in the sentence. Other is sustitutional, it replaces the set of things in focus with a new thing or set. The funny behavior with respect to determiners only applies to a/an, and is simply a way to make sure we don't mistake an other for another, since they are completely different in terms of sentence meaning. It has nothing to do with word choice. You don't choose between other and another based on determiner use or noun types, we merely avoid any potentially ambiguous construction for clarity's sake. Thus some other, the other or any other is used in stead of an other.
In the not so distant past, the construction "or another" was nearly nonexistant in English. It makes no sense to use a concatenative word with or. I still don't ever say "or another" - it has to be "or some other" or just "or other".
While Ngrams now shows "or another" to be common, and "one or another" (which I can't even say) to be more common than "one or other", historically, they were rare.
The most likely case at hand is that the definition was written down at a time and place when "or another" simply wasn't a grammatical option. And there are still some of us for whom it isn't.