In your lesson you say that demeurer, when used in the perfect tense with avoir or être, behaves in the opposite way to other two meaning verbs. Is that right? Does it not behave in the same way, ie. it takes être when intransitive and avoir when transitive?
Sorry, perhaps this is not right. For example, il a demeuré à Paris is an example of intransitive avoir use.
So, is the rule that we use être when the usage is intransitive and expresses a state of being?