To my knowledge the question of whether Gandalf actually wore Narya is uncertain. All that we know, as you have already pointed out, is that he didn't wear it openly.
This thread discusses the meaning of Gandalf's words to the Balrog in some depth. Briefly put, the 'Secret Fire' is the Flame Imperishable, which is in Tolkien's words the 'creative activity' of Eru - the part of Eru that gives being to his creation. I would interpret this as Gandalf stating his allegiance to Eru.
The meaning of 'Flame of Anor' is unclear. It is known that Anor means Sun, of course, but this doesn't help us a great deal. Some have suggested that the phrase refers to Narya. The aforementioned thread gives arguments both for and against this.
[ December 29, 2002: Message edited by: Voronwe ]