I am not offended by the traditional use of the term "lady" at all - in fact, I actually kind of like the old-fashioned definition (at least how I interpret it) as a woman who is classy and treats others politely and with respect.
However, the more modern use of it among the younger generations - well, it doesn't inherently offend me but it just makes me squick: girls in their teens and twenties wearing short, tight, slu*ty clothes, painting themselves to look like sparkly strangers, going out drunkenly on the pull in grotty clubs and then posting photos of themselves on facebook with the squealy caption, "Ladies' Night Out!! !" In my mind, those aren't ladies. So if anyone referred to me as a lady in that context I wouldn't just be offended, I'd be furious. Unfortunately, now, I am more likely to think of this new definition of a "lady", and often imagine the word being used by skeevy players eying up underage girls too foolish to know what they're doing. Ugh.
And just a note: I think this discussion was meant in the spirit of discovering how different people feel about the word, not about deciding if one group or the other was right in their opinions. Perhaps I'm wrong?