Is this the result of Medieval scholars editing the text to coincide with Christian ideas? Is it supposed to be understood that they mean Zeus or Jupiter, since he is the most powerful god in the entire pantheon? Is it a more philosophical idea of divinity, e.g. logos?
It's hard to gauge authorial intent without specifics, but on the question of whether the worship of one God existed in the Greco-Roman world outside of the Judeo-Christian tradition, the answer is 'sort of, but it's complicated.'
In the Eastern parts of the Empire from circa 400 BC there emerged 'Hypsistarians' who were cults worshipping 'Theos Hypsistos' (The most high God) these weren't monotheistic (the belief that only one God exists) but rather they were henotheistic (worshipping one God but recognising the existence or the possibility of the existence of others)
The term 'Theos Hypsistos' could refer to Zeus or Apollo or other Greco Roman deities, it was also used in areas where interaction between Jewish Groups and the Greco-Roman world occurred to refer to the Judeo-Christian God.
The fact that the 'Hypsistarians' were located in areas with exposure to Judaism suggests that Jewish Monotheism was a strong influence in its development.
http://www.academia.edu/1462334/The_Cult_of_Theos_Hypsistos_in_Roman_Thessalonica_in_English_
Honestly, it depends on the context. In Plato's Apology, for example, "Θεοσ" Socrates addresses is Apollo, which is all around fairly common in the Platonic corpus. Seneca, on the other hand, is generally referring to a somewhat nebulous natural pantheistic conception of the divine when he says "Deus".
To complement your other answers, some Stoics believed in one god but the worship of the many different aspects of him under different guises.
[A] portion of him [God] pervades everything, which is called by different names, according to its powers; for they call him Dia as being the person (di hon) everything is, and Zeus, inasmuch as he is the cause of life, (tou Zēin), or because he pervades life. And Athene, with reference to the extension of his dominant power over the aether (eis aithera). And Hera, on account of his extension through the air (eis aera). And Hephaistos, on account of his pervading fire, which is the chief instrument of art; and Poseidon, as pervading moisture, and Demeter, as pervading the earth (Gē). And in the same way, regarding some other of his peculiar attributes, they have given him other names.
Diogenes Laertes, Life of Zeno
I suppose this should fall under /u/intangible-tangerine ‘s ‘it’s complicated’ label. Despite this general idea as explained by Diogenes Laertes, we know other Stoics seemed to have the belief in many individual gods. Marcus Aurelius in his Meditations makes this assumption, and even that these individual gods could be communicated with.