...Or did poetry<->music operate on a continuum that could depend even on a particular performer or performance?
Naturally I'm interested in the medieval angle, too, but I don't want to "since the dawn of time" it here.
Arma virumque cano is the opening phrase of the Aeneid; it means "I sing of arms, and of the man..."
Sort of, your third point is probably the closest. In many ways the closest translation of "cano" is its etymological descendant, "chant". If you want a good idea of what this would be like, there are certain anthropological parallels that can be found today, a particularly well studied example being the Akyns of Kyrgstan, particularly the great Manas Epic Trilogy. If you watch those clips, you can see a form of poetic recital that doesn't really fit into modern divisions between poetry and song, and while of course the Krygyz tradition is not related to classical Mediterranean poetry it gives an idea of how these would be recited.
Now of course there is one major flaw in this: Kyrgyz epic poetry is oral, only recently being transcribed, while Vergil was very much a literate author. That being said, Vergil's poetry was in conscious imitation of the "rhapsode" tradition from pre-Classical Greece. So it isn't a perfect comparison, but again it gives an idea.
I'll focus on the ancient Greeks and Romans for this answer. Even with this focus, however, there's still some variance in the connection between poetry and music. This variance was somewhat dependent on the performers, but it was mostly dependent on what was just common practice at various points in time.
The text of the Iliad and the Odyssey indicate that these epics were initially performed by singers accompanying themselves with a musical instrument (these performers were called aoidos by Homer), but during the classical period these pieces were simply recited by performers. Hesiod's works, like Homer's, were also initially performed with the performer accompanying themselves with an instrument. So, both Homer and Hesiod were writing their epics during a time when epics were commonly sung and accompanied by musical instruments (In fact, Pausanias noted the both Homer and Hesiod participated in kitharode competitions). It appears that from around the sixth century B.C.E. and onward, however, the performance of epics shifted from singing with accompaniment to recitation. Scholars believe that this shift may have something to do with how the epics were transmitted and with the growth in audience size.
Jumping ahead to Rome:
Virgil wrote his epics at a point in time where epics were still mostly unaccompanied. The Romans believed that a poetic text could be considered complete even without accompaniment. However, in Nero's time Virgil's poetry was sometimes accompanied. For example, portions of the Aeneid were sometimes given musical performances. The singing of Virgil's poetry, however, mostly relied on the natural rhythms and syllabic patterns of the poetry.
So, most Greek and Roman accounts of performances indicate that, with some exceptions for the early Greeks, epics were typically recited without instrumental accompaniment. When Romans did accompany recitations with an instrument, it was typically only for excerpts of epic poetry. Answering your question is tough however, as I don't want to speculate on what Greeks and Romans considered "music." I can say, though, that the recitations of epics would surely fit into the Greek concept of "melos," as it had musical elements in the singing, and used the text's inherent elements of rhythm.
Sources:
Grove Music Online articles for "Homer," "Hesiod," "Aoidos," and "Virgil," and contemporary accounts mentioned in those articles.