[mjb]

Exactly.

CAP is one of the borders of the space. It's popular because it's relatively easy to look at a claim and see if it's on the "possible" or "impossible" side of the border. Tools like that (as in your halting problem example) are extremely useful, because they can be used to cut through a lot of bullshit masquerading as subtlety.

On the other hand, as OP's post does a good job of explaining, CAP is not a good tool for exploring the area inside the boundary. We often use it to do that, and it leads to all sorts of awkwardness and miscommunication. I think OP captured that well, and it's an area I'd really like to see some alternatives getting popularized (harvest and yield, PACELC, etc). Simplified rules are useful in this space, because while it would be very useful for all practitioners to understand the whole taxonomy, there's a lot of material to understand there.