I'd argue that higher management is usually accountable, but not responsible. Like, they can't guarantee fulfillment of the individual pieces of the work, but are still accountable for the whole effort anyway.
(this is going by the dictionary definition that responsibility is "task-oriented")
> accountable 1. (of a person, organization, or institution) required or expected to justify actions or decisions; responsible. "parents could be held accountable for their children's actions"
So, it seems to mean you can justify your decision, which relates very directly to what the parent was talking about. You face punishment if, when called to account, you cannot explain what you did and why in a way that justifies your actions.
In my opinion, their description of the distinction made a lot of sense and provided me with a valuable tool for separating the two concepts.
edit: as I've thought about this more, I do see that the google quote literally includes the word "responsible" inside the definition of accountable, which goes against my point. I still really feel the distinction, especially the with the example of a person responsible but not accountable, seems sound and useful.
> So, it seems to mean you can justify your decision
No, not at all. If you can justify your decision, that doesn't mean you're being held accountable for it. Being held accountable means being punished, whether you can justify a decision or not. If you cannot be punished for something, then you are not accountable for it.
That's not how I would understand the word. It also isn't how dictionaries understand it.
Accountability means being subject to punishment. (That is, you can be "called to account".) It is, technically, different from responsibility - the word for someone who has accountability, but not responsibility, is scapegoat.
However, it is generally agreed that operating with scapegoats is both morally bad and bad for productivity (and the same goes for giving people responsibility without accountability), so responsibility and accountability aren't supposed to be separate concepts.