椅子の上にあるテレビがついています。 The TV on the chair is on.
That's basically correct, but notice that in a sentence like this one, certainly *somebody* turned the TV on. The intransitive is sometimes used when we just aren't interested in specifying who did that.
In ドアが閉まりました, it could be any of:
a) Some unnamed person closed the door.
b) Some natural force (like wind) closed the door.
c) It's an automatic door and literally closed itself.
In this kind of sentence, it's not there's necessarily no actor, it's that we are only trying to convey that the door closed (or that the TV is still on, etc.), and aren't interested in conveying how that happened. If it's important to clarify that we really do mean the door closed itself we can say something like ドアが自動的に閉まりました。
Or consider 僕は6時前起きました for example. The same event could possibly be described as 六時前母が僕を起こしました, or possibly it can't, maybe the speaker woke up on their own. In the first sentence we just don't know.
This isn't really any different from English. If someone says they got up before six, it presumably doesn't matter if they woke up by themselves, an alarm clock woke them, or a person woke them. If it did matter, they could have been more specific.
So, yes, the intransitive says that 'subject did an action', and it often means the subject acted on their own, but not necessarily.