I do agree with your point about hidden and accidental intentions.

How to decide whether a picture that wasn't intended to have a 'meaning' has one is simple. As soon as a meaning is discovered, you'll know it.
The question whether the person who unwittingly put that in what was intended to be just a pretty thing is an artist is another matter. Perhaps, in a way. Perhaps not.

An undiscovered meaning, put by an artist into his art, is a bit different.
He or she may be a great artist, potentially, with important things to share with us. But not a great communicator.

It's like wanting to, say, warn the world about impending doom. Being manically driven, perhaps, to save mankind by warning us of great perils that lie in our immediate future.
But doing so in a book written in a language of your own invention that nobody understands, and putting that book on the shelves of libraries expecting that that would be the salvation of humanity.
Might that book be legitimately called a warning to us all?

A great depth of meaning is not there, if there is no way anybody can find it. So an artist that intends something that nobody perceives in his or her works may be a great thinker, potentially a great artist too, but definitely not yet.

You (again) put this "what do i care what you think" thing forward.
It rather is a "what do you think?" thing.
A thing with a quesion mark at the end. We (i at least) do care what you think. Else there is no point in all of this (not just this discussion alone, but art also).

But i do agree that if you find a deeper subject, some meaning, some relevance for you, you do indeed do just that. Even if i don't.
I never said that it need be a universal thing. Some people like to talk about motor cars a lot, and can get very passionate about some obscure difference between some equally obscure part burried away somewhere in among many other obscure parts. Great!
But not my cup of tea. They might as well be talking in Mandarin for as far as i can understand and care.
Yet that is not part of any consideration about whether it makes sense or not, whether that talk about these bits has a proper subject matter. (At least not part of any of my considerations).

As for the 'definition' of fine art being narrow. The important question is of course whether it needs to be wider? Is it not comprehensive?