Let me ask you a question though, can you be swayed to alter your position? I don't know if you are or aren't, but sometimes I think we forget how stubborn we can be holding onto our own worldview, and then apply a different rule set to an atheist. For me, I generally try to concede a point if I truly believe it has validity. I suspect however, their refusal is no different than anyone else with a worldview.
I think a distinction should be made between arguing for theism / deism, or arguing for Christianity. I think many atheists would have an easier time rationalizing, say deism, over Christendom. The move to Christendom would most likely require "faith" as it were.
I'm not quite sure what you mean by alter my position.
There is indeed an entirely different sort of discussion when you are engaged in an inter-faith dialogue-- Christian to Jew, Muslim, Buddhist, etc., versus a theist vs. atheist discussion. The strain of American atheists who like to argue (which I suspect is a minority subset of the whole population thereof) usually fine-tune their points to focus on either Christ specifically or the Abrahamic God more generally, since those are the by far the predominant religions in the U.S.
But they also like to respond to any assertions of God's existence by saying "Which god?" and go on to say there is absolutely no difference between Christ, Yahweh, Allah, and Zeus or Thor or Quezacoatal (sp) or (their favorite for illustrating absurdity) The Flying Spaghetti Monster, because all have equal evidence of existence, and that is exactly zero evidence, in their view.
And then they'll go on to say how naive it is to just decide you're going to believe in something and base your life on that, and how it's childish to base your actions on gaining either eternal reward or eternal punishment, and that belief in any kind of higher power is not necessary to have a strong moral compass (that one I kind of agree with, to a point).
These arguments are unwinnable, as both sides are convinced they're correct and they're not going to change the mind of the other.
Now, if the discussion is with an agnostic, someone who says they just can't decide for sure whether or not God exists, then you are going to be drawn into the paradox of omni-God, that is, how can God be omnipotent, omniscient, and omnibenevolent all at the same time. I admit that this is where I still struggle, because ideas like unconditional election and limited atonement horrify me. They suggest that God creates human souls, sentient beings in His own image, who were damned from the beginning of time. I'm not the only one who can't reconcile that thinking with the idea of God being all-good. But that is a discussion for another thread.