[Keeper of Dol Guldur]

See that's where I'd tend to disagree; somebody said in another forum; "Why would Gandalf reveal that to a mortal enemy?"

I mean, if Gandalf got killed by the Balrog, it seemed likely that the orcs would desecrate his body and find the Ring of Fire, and eventually word would reach Sauron that it had been kept by Gandalf. If he was going to be dead, maybe he didn't think it mattered if anybody knew, but it doesn't seem very much like Gandalf to go announcing something like that. Especially since the bearers of the three had some ability to 'see eachother from afar'. If Sauron got his hands on Narya, he'd know Elrond's and Galadriel's locations, strengths, and plans. They had to remain hidden, which prevented Gandalf from openly saying such things.

On the other hand, he didn't just blab; "wielder of Narya, Ring of Fire." He could have used 'Flame of Anor' just to illustrate to the Balrog that he was in fact wielding a weapon which could be used against the shadow of the demon of might. That and it's likely that the Balrog either had no clue of the existence of 'Rings of Power' or else it just didn't care. Obviously it was okay if he in doing so let it slip to the Fellowship - he trusted Aragorn to maintain the secrecy of his keeping it.

I still say Flame of Anor is comparable to 'Light of Day', because Anor (Anor, Anorien, Minas Anor) is very specifically the sun, and Gandalf lit up a lot of different things. Naturally, the flame elves would place into a ring of power would be that of the sun, because they didn't love the tumultous fires of the earth. Those fires are naturally having to do with the One Ring, not the three.

As for "Servant of the Secret Fire", yes that seems to be referring to his status as a Maiar. He's letting the Balrog know that he is its equal in strength, and that he has a weapon of light also, the very power of it is not to destroy, but it's make is something that in itself would hurt a creature of shadow - sunlight.

Gandalf really liked to let people know it when they assumed they were more powerful than him. The Balrog . . . Saruman . . . Denethor . . . The Morgul-Lord . . . he liked to put things in their place.