Just to clarify, does that mean that what kake referred to as "third stomach tripe" is "omasum"?

Yes, that's right — the type of tripe used in this dish is from the omasum.

What is the "honeycomb" tripe? Is that a specific chamber as well?

Honeycomb tripe is from the second chamber, the reticulum. I mention this towards the end of the post, though I see when I moved that paragraph I should also have moved the mention of the reticulum a bit earlier. I will have a bash at making it more readable later on.