If we've learned anything about what is genuine and what is not, our actions dictate how genuine it really is. She came to you crying and desparate about getting back together and how sad and miserable and lonely she is. Do you really want to base this new relationship (it better not be "getting back together" you are insane if you think that getting back together will end in you two being happily ever after) on how she wants you because she NEEDS you to get out of this pain and sadness and unhappiness. Do you really want somebody that needs somebody (don't talk yourself up here, you aren't the only "one" for her) to live her life?
I know you want to follow your heart, but you have to understand that emotions can run our decisions and without logic. Did you feel terrible and sad and pity for her as you were taking her back? You really think that when she tells you "I've changed, I'm grown up, I'm a different person" that she really has? She may think she has but how much can somebody really grow up and change when she chased another guy, it failed, and now wants you back? Where was the time and effort put in to bring about this change? I'm telling you right now, the people that have truly grown up and truly have changed, you can tell in their actions, how they speak, everything. Anybody that has to say "I'm changed" feels the need to prove something, and if you really have changed, you won't need to prove it.
I don't want to say I told you so, and I really hope it does work out for you. Now is the time to really go in depth on what went wrong and what behaviors you both have done to break it the first time. There was a break down in communication. There wasn't any honesty in your feelings. This should be discussed and brought out before you fall for her again and even when you do, you have to be on your guard and pay attention to everything she says and does. It sounds exhausting but as I've learned, relationships are consistant work.
Waking up next to a beautiful girl,
Step outside and say hello to my beautiful world.