J.W.
Oh T., I have been there many times. I really like Catherine's advice, but wanted you to know that I, too, have one of those relationships with my daughter where we can be best of friends and wosrt of enemies.
In my experience, I found that it was me who had to make the changes. My first instinct is to blurt out something not so nice, then how can my daughter not reply the same or not be hurt. They are at such critical development ages between 7ish and 16. There is so much out there in ads, school, etc. which can damage their self esteem while they struggle to find out who they are.
I don't know if you ever saw, Nanny McPhee, but someone suggested I start doing what she does in the movie. If I saw something or heard something I didn't like, I go "Hmmm" and hold my tounge. My daughter knows I've seen or heard something I didn't like and the pause allows her to process it while I calm down and find a good way to present it.
She needs discipline, but also love and positive support. When it comes time to talk with her, if it is appropriate, talk about some of the things she has done well and give her praise. Then talk about the behavior/words you need to address, then leave it on a positive note or a plan of action.
I know changing can be hard, I still have to remind myself when I get frustrated. I have to go to my room and give myself a time out sometimes to gather my thoughts and calm myself. Know that in this current world, the home should be the place of consistent unconditional love and support.
As I reflect on the biking outing, I wonder if the fact that her friend was riding better than her was a bit intimidating. Perhaps everyone taking a break or everyone walking with their bikes so she can rest up...or ask her what she would like to do to finish the journey she wanted to start...she evidently saw an opportunity to spend time with you, but someone was riding better than her and perhaps the path was too difficult for her.
Anyway, I don't know if my experience will help you, but know that I've been there and it is a stuggle for me to compose myself on a lot of days. But since I have done so, things have improved with my daughter....she seems happier and it comes through in her eyes and her behavior.
Good luck!