That's true. I added this motive for the sake of completeness.

It seems to me that if the Balrog was being drawn by the Ring that doesn't mean it knew it was being drawn by the Ring, it may have simply been compelled to follow this urging.

I also feel as if its main reason for attacking the Fellowship having been drawn to them by the Ring may well have been that same kind of nihilistic hatred we are told Morgoth inspired in his servants generally (to varying degrees). Might it have attacked simply because, in its very evil and corrupted state, it simply felt homicidal antipathy for Men and Elves (and, evidently, Dwarves)?

Sure, it may well have been a decision based on curiosity rather than the clear intention to get a hold on the ring of power to archieve some kind of goal.
As a side note: I wonder how exactly the Balrog perceived Frodo. Did the Balrog perceive Frodo as the little, round and ordinary Hobbit, that he is (in a strictly physical way) or was there more to it?! The Orcs at Cirith Ungol mistook Sam for a fearsome and mighty warrior because he had the Ring. I can't remember if Sam was actually wearing the Ring on his finger at this point or not. Furthermore we have Gandalf's notion that Frodo's "appearrance" (or Aura?) wasn't the same anymore (depended on the observer) after he got stabbed by the Morgul Blade.

I think the Balrog would have found the hobbit very interesting. I think it could sense that he had been in pain prior, but had a strong enough will to continue. Being able to look at people and immediately understand these things about them seems to not be uncommon among the Maiar.