Is "capability-based security" vs "hierarchical-protection-domain based security" a false dichotomy?
No.  I don't think that would be a good summary, as it doesn't get to the heart of what I consider important about capability-based systems.  Your rephrasing is putting a focus on how the restrictions are enforced, but that is not the interesting part of capability-based models; the more interesting part is how we express our security goals as capabilities, and what advantages and disadvantages such an approach has.

First off, understand that the notion of capability-based security is not a precisely well-defined concept; it's a broad phrase that could cover multiple possible meanings. Different people use that phrase to mean different things.  So take everything here with a grain of salt.

I suggest that the best way to understand the phrase "capability-based security" is "a security architecture that relies heavily on capabilities".  So, then you need to understand what capabilities are, and why they might be useful.

If you keep reading that Wikipedia article, it contains some explanation of what capabilities are and some ways that they might be useful.

If you want to dive into the details of one particular view on capabilities, I recommend looking at https://en.wikipedia.org/wiki/Object-capability_model and Mark Miller's PhD thesis.  I think you'll find that to have a coherent, principled, and focused view.  It's not necessarily what everyone means when they use the word "capability", but it's one take on the subject that is internally consistent and interesting.