Salut -
In the kwiziq glossary entry for Verbes aux deux auxiliares, it mentions that demeurer follows "the reverse pattern" to the transitive/intransitive rule.
I have used that as my rule of thumb for a while now, and it was always confusing to me (as a verbe d'état, it is intransitive, yet takes auxiliary « être », so that note didn't feel right).
I just found the comments and explanations here on this page, and all is now clear - thank you Aurélie and other contributors for the information you have shared here!
It might be worth correcting that glossary page entry, too, in case others stumble across it in the future.