First, I want to tell you that I think it is very courageous to be willing to put yourself in a vulnerable situation by admitting your mistake and requesting a renewal of the friendship. I just wanted to acknowledge that it is one thing to know you "messed up" and wished it was different, and quite another to be willing to actually DO something about it. So, yea for you! No matter what your friend chooses to do when you contact her, you have taken a great, bold, courageous step for yourself.
I personally think it matters little how you choose to contact her. The main thing is that you do. Sometimes, we stumble around in our attempts to do things perfect, when doing it is the most important thing. We can clean it up as we go through it. No matter how you choose to contact her, there will be some awkwardness at first, some hesitation, some level of self protection. Yet, as you are willing to be honest and authentic with her, the trust between you will grow, and you will be able to get to the place you want. It may take time, and likely will not happen instantly. I encourage you to allow you both to have time to process whatever feelings come up and give each other lots of compassion. Trust in the relationship that you had before and believe in it's power to heal the current divide.
I can say this both from a professional standpoint as a certified life coach, and from personal experience as I not long ago healed a very close friendship of 25 years that was estranged for several years. I can tell you that I am so grateful that we were able to heal our differences, forgive, and reconnect. Our friendship is actually stronger and closer now, which I never thought was possible when the unthinkable rift came between us.
I wish you all the best in your courageous pursuit!