I was going to post to this yesterday but I was getting kinda frustrated reading it. This is my honest opinion: You had her from the beginning, but you screwed up. Not because you went to a family party, not because you didn't "take her away" from the nag, but because you are WAY too damn wound up and on edge, and worried about every single detail.
From the beginning she was into you. She gave you her number, she invited you to a cookout. Everything was fine. But you were SO concerned that you did something wrong just because she didn't immediately shove her tongue down your throat, or accept your date offer when she had something else to do.
It's almost as if you EXPECT this to not work out because you perceive her to be SO incredible. And since things haven't progressed exactly according to *your* internal agenda, you're getting all bent out of shape and assuming that anything she does "means" something. Ex: "I can't go out on Friday, I have to work" you interpret as "She doesn't like me, I must have screwed up". Whereas if it was just a friend saying that to you (guy or girl) you wouldn't think twice about it and just make plans for another time. So what's the difference? Why is it the "hotter" she is, the more she is probably lying or "flaking" when she can't make a date? I would explore that if I were you. Even if you're not guilty of the above example specifically, you are misintrepreting her actions.
It's as if you're saying internally: "I could never get her, she is just too incredible and glows to me. She'd never want *me*." And now, everything she does you're warping and reshaping in your head to mean exactly the above. To prove yourself right.
Even now, you probably have a shot. But you have to relax. Put it this way, she either wants you, or she doesn't. If she does, you won't have to run any kind of 'game' on her. If she doesn't, then there is no 'game' in the world that will change her mind.
I'm not sure *why* you can only go out in 'group settings'. And if that is the case I really don't know how you will ever be able to talk one-on-one with her to connect with her and truely get to know her. So that part you're going to have to figure out yourself. I think it's silly that at 21 and 18 years old you can't go out to eat together by yourselves. But hey, who am I?
And you won't have to "take her" from the other guy. Like an above poster said, he never had her. And even if he did, she would have to CHOOSE to leave him anyway, you wouldn't be taking anybody. So just relax, call her, talk to her, get to know her. I'm pretty sure the 'forgetting the name' thing was part of her game. That's ok, let her play if she wants. You just be real and honest and she'll eventually see that she can trust you and let her guard down.
Good Luck.
Blitz