I am betting you know what mandatory access control is ; ). They basically amount to a firewall that is placed on applications restricting what they can do. The rules are generally written by downstream distros and are divorced from the implementation. The problem is that it's hidden control flow, so the program just dies and can't fall back gracefully. Capability oriented APIs make broker processes and narrowing of permissions tractable.
[1]: https://sel4.systems/performance.html
[2]: https://lionsos.org/
[3]: https://trustworthy.systems/publications/papers/Heiser_25%3A...
through what exactly people mean with it is often vague
Like e.g. both seLinux and AppAmore are technically MAC but people tend to only mention seLinux when speaking about how cumbersome it is and treat AppAmore as something different as it's not so cumbersome.
I'm bullish on capabilities too, but I don't know much about MAC. Can you explain your last sentence?