Not ready for a relationship but acting like it!?

Ive been seeing this girl for a little over 4 months now and she has indicated from very early on that she does not want a relationship because she had just gotten out of a LTR 1 month before we started dating (of 3 years). As I am fresh out of a LTR myself, I agreed and am on the same boat. We have an understanding.

A little history. We used to work together for about 2 years and have always stayed friends and hung out in the same circles. We both always had a significant other so nothing ever happened, no flirting or anything. There was always that level of respect there. However, after we did start dating we confessed that we had secret crushes on each other

We kissed about 3 dates in and had sex after about 2 months. All throughout we have do everything that couples do: hold hands, text all day and night, kiss, have sex, go out 2-3 times per week, she stays over at my house at least once a week, cuddle, cook dinner, go shopping, concerts, dinners, movies, the whole deal. The only difference is we dont go out or meet each others friends or family and do not introduce or refer to ourselves as each others bf/gf. She has indicated to me that I am the only person she is seeing and I have told her the same. But we havent had the talk about exclusivity because of "our arrangement."

Here's the issue: Unfortunately, I have fallen for this girl and would like to call her my girlfriend and maybe attempt a relationship. However, when we do talk about the topic she gets distant and reaffirms that she is not ready. At this point I dont know if Im ready, but I am a risk taker and would like to find out. I dont want to push her away but also dont want to loose the chance to have something great. I guess my confusion is whats the difference between what we are currently doing and an "official relationship?" Should I just leave the issue alone and let things take its course? Should I bail--- is this girl too damaged? Part of me wants to appreciate the situation but I am a one-woman kind of guy that wants to be sensitive to her feelings... But also wants to know we are on the same page to avoid getting hurt myself...