I believe "archaeal cell" is referring to an Archaea, one of the three branches of life. All three branches derive from a more distant ancestor, LUCA. LUCA was undoubtedly preceded by other ancestors, but there is (by definition) nothing else branching from them that has survived.

I anticipate the definition will become increasingly subjective as we find biology-messiness inconsistent with our concepts of ancestry.

For example, suppose horizontal gene transfer occurs from organism X to organism Y. Does that mean Y is now a branch of X?

* Does it depend on how much was transferred?

* Does it matter only if the specific sequence was passed down? If so, how much mutation is too much mutation?

* What if the same end-result occurred through a retrovirus instead of a plasmid. Is the virus an ancestor too?

* What if the swap was simultaneous and bidirectional?

* What about transitive links to organisms W, V, U that did the same?

* Are mitochondria "us" yet? If so, are we the ancestors if they redevelop enough machinery to "escape"?

etc.