me: A "p-zombie" "behaves" the same way we do, but does a p-zombie believe it has qualitative awareness?

To be precise about the value of the belief/intuition concept in accounting for the illusion that qualia exist—one defect in the zombie thought experiment is that it prompts the attitude: maybe I can't prove that you're not a zombie, but I sure as hell know I'm not one!

The zombie experiment imposes a consistent outside view; it seems to deny the evidence of "personal experience" by fiat—because it simply doesn't address what it would feel like to be a zombie.

So, the zombie experiment seems to show that people might not be able to distinguish zombies from humans; but invoking the beliefs held by the "zombie" shows from the inside that being a zombie can be no different from being a human: the two are subjectively indistinguishable.

To address your question directly: the ordinary zombie thought experiments purport to show that without qualia humans would be zombies; whereas when you allow zombies' (false) beliefs (in ineffable perceptual essences), the thought experiment shows that zombies are really humans.