Where, may I ask, do you find "authoritative" support for the conclusion that only services performed while the tenant occupies the unit are relevant to this determination? The "highest" authority that I can find (the IRC 1402 regulation that I quoted earlier) doesn't appear to make that distinction. Rather, it simply speaks of services rendered "for the convenience of" the occupant.
So, I do think there is a distinction between the two duplex rental scenarios you presented. In the case of the "furnished" unit, I believe a "facts and circumstances" inquiry would be called for to determine whether or not services were rendered for the convenience of the occupant (see reg for exact language).
But keep in mind, I'm perfectly open to being persuaded differently. All that would be needed is an "authoritative" source that leads to a different conclusion than the one I've stated above. ;-)
(Note to non-pros who might be reading this: "Authoritative" is a term of art. It means a source that can be cited in court as precedent. IRS publications, articles by H&R Block, opinions of tax preparers (including myself!), etc., are NOT in and of themselves "authoritative.")