I guess there are different kinds of nomenclature for these things. A long time ago, I was taught that there is sometimes an expletive. I think I was, in one English class or another, even given expletive as a part of speech. It struck me as strange, because the word expletive had by then come to mean “dirty word” (because of its use during the Watergate scandal).
This Wikipedia article uses the term “syntactic expletive,” but it also uses the term “dummy subject” (the term used in this thread). I was taught that there wasn’t the subject in such sentences. In sentence diagrams, I forget where we put it (I’ve forgotten how to diagram sentences), but I’m pretty sure we didn’t put it in the subject’s place. ( But I (re)learned sentence diagramming (in college) from an old book.) On the other hand, we were taught that, when it was used that way, it was not an adverb. However, I think that there can sometimes be used adverbially to introduce a clause, as in examples I gave earlier in the thread (Coleridge: "And there were gardens . . . [a]nd here were forests . . . "; P.G.T. Beauregard: “There stands Jackson . . .”).
I think I’ve even seen there (the there that is sometimes called a dummy subject or a syntactic expletive) described in a respectable grammar book as an adverb (and I think they knew what they were doing by doing so; I think they made a conscious decision to “deem” or “dub” it an adverb).
I’m not quarreling, or even quibbling with anyone. Nomenclature and concepts differ from place to place, and they change over time. I think it’s a good thing to have some kind of nomenclature, though, but in grammar, differences in names or even concepts don’t matter too much to me, as long as I have the opportunity to get on the same page as the person using them.