[Man-of-the-Wold]

I'd confirm all that's been said, and just add that in a moment of confrontation with the Balrog, Gandalf should not be expected to be pedantically precise with his expressions.

In general he is at that moment drawing on his linkage to all that is good and powerful, right to Ilúvatar, to whom through Manwe he is faithful as Olórin. So, it is not about the Ring of Fire or his regular Wizardly ways, by which his Maiarian origins are usually suppressed.

The "flame of Anor" is, perhaps, some spur of the moment piece of poetic license to punctuate the powerful reference to the Secret Fire, which the Balrog cannot overcome; the bottom-line being that you will not get to the Ringbearer! Anor is a word for the Sun at that time (what else, who knows for sure?) But, through Laurelin the Sun is derived from the power of the Ainur, which comes from the Flame Imperishable and thus, Ilúvatar whom they serve.

Also, Westron may not have offered any better references by which Gandalf could make his command and convey his will at the Balrog. It did not seem that Middle-Earthers had a very developed understanding of the Metaphysics behind their World.

The flame of Udûn is a case in point. He is essentially calling the Balrog a Fire of Hell. As a fire demon this is figuratively indisputable. But we really never get much information at all about what "hell" for Arda might be, beyond Melkor as Lucifer-like, and his fortresses not unlike the Underworld. Indeed, the story of Beren and Lúthien is not without echoes of Orpheus. How the Balrogs might fully relate to any such notion of Hell is just a mystery of Tolkien's imagination.

Also, to use the Bible is always a sound way to understand any number of writers, including many who were never anywhere near as pious as Tolkien. One does not need to have fully structured allegories to be greatly influenced by the majesty of the Bible. Both Gandalf and Elrond (to name only two characters) seem reminiscent at times of Hebraic, Hellenistic, Apostolic and Saintly forebears or perpetuators of Early Christianity, as opposed to pagan-like mystics with which they might superficially be compared.

Indeed, Tolkien was in many ways trying to cast pre-Christian (or at least pre-Norman and pre-Carolingian) images and traditions from Northern Europe in a light that were fully amenable to Judeo-Christian ethics and values. A real legendary base for the English.

In summary, I think that with Gandalf and the Balrog, as well as in other places in The Lord of the Rings (e.g., Eye of Sauron, the Redhorn, the Sun on Mount Mindolluin,) fire stands for good or evil in ways not unlike Biblical verses. I can even think of a few Burning (albeit consumed) Bushes.