Is this looking through the frame work of a Christian God in early translation efforts (in the same way 'thou' and 'canst' slip in), is it Bertrands interpretation of ambiguous terminology, or is it part and parcel of theological philosophy of the time where God as truth (rather than personal) was held in equal likelihood or value as polytheistic personal gods or monotheistic animal gods.
Edit: figures in the pre-socratic era such as Parmenidies, Anaxagoras, Protagoras, and even Socrates himself.
This is very much a feature of older translations of ancient texts. As /u/mizzamir points out, the Greeks made frequent reference to "the god" (ho theos), which usually meant either Zeus or Apollo, depending on context; the addition of the article shows that this was not an abstract concept of absolute divinity (capital G God), but a specific deity invoked for the occasion. However, this appears to have been a bit too ambiguous for early modern translators working in a very Christian world, so they often simply rendered ho theos as "God".
Similarly, when the Greeks describe things as coming from a god or being "of the gods", older translations often render this as the more neutral "from Heaven", which is not a concept the Greeks would have recognised.
There's a great example of this in the 1921 Loeb Classical Library translation of Xenophon's Hellenika. The text (4.4.12) runs like this:
ἔδωκε γὰρ τότε γε ὁ θεὸς (the god, i.e. Zeus) αὐτοῖς ἔργον οἷον οὐδ᾽ ηὔξαντό ποτ᾽ ἄν. (...) πῶς οὐκ ἄν τις θεῖον (of the gods/divine) ἡγήσαιτο;
The translation, however, goes like this:
then at least heaven granted them an achievement such as they could never even have prayed for (...) how could one help regarding this as a gift from heaven?
I don't know if Bertrand Russell knew Greek, but even if he did, he may not have cared to correct the anachronism in the translation he cited.
Ancient people referring to specific gods, for reasons I don't think are entirely clear, tended to not refer to them by name. Whether this was a taboo or simply a standard in writing, I'm not sure.
However, when Socrates referred to "the god," he meant the god he believed had intended for him to be a philosopher. That's the god of Delphi, Apollo. There's no evidence that he was ever anything but totally sincere about his piety with regard to Apollo. Despite what a reprobate like Meletus would have you think.
We see a lot of references to "the god" or "the goddess" in ancient texts. The context and source would have made the particular member of the pantheon being referred to readily apparent to a contemporary. Homer did it too, and generally translations will either put in the god's name or just footnote it.