I am in graduate school to become a marriage and family therapist and I'm the mom of a 2 yr. old girl and a 4 yr. old boy. I've been learning some stuff that has helped me a lot. A saying that's opt quoted at school is: A misbehaving child is a discouraged child. Your daughter clearly sounds discouraged. Could it be your six year old is keeping you engaged with her by misbehaving? After all, she has your full attention now, you are consumed with what to do about her. Sibling rivalry is a common occurrence. Your older daughter was unsurped from her special position when the younger was born and now the younger one is even catching up to her skill-wise. If your child is misbehaving for attention, then her underlying mistaken belief is "I only count when I'm being noticed." She feels insecure and alientated. Her negative goal is attention. If this is the case you will feel annoyed and irritated. Another possible motive for her behavior is POWER. If this is the case she will believe that her strength is in showing you that "you can't make me" or "you can't stop me". Her defiance indicates that this may be the case. In addition, if power is the motive you will feel angry and challenged, which does seem to be what you are describing. You impulse in this situation is to fight: I insist that you do as I say. The misbehavior then intensifies. A third possibility is REVENGE. If this is the case she believes "I knew that you were against me. No one really likes me. I'll show you how it feels." She feels insignificant and her goal is to get back, get even. You will feel like you want to hurt or to punish her, which sounds like what you are describing. Your thoughts may be, "How could you do this your sister? I'll teach you a lesson" (by the way, I'm getting this right off a handout by Amy Lew, Ph.D. and Betty Lou Bettner, Ph.D.). Your daughters reaction to your correction will be that she wants to get even and make self disliked. What she really needs in this situation is to feel like she COUNTS. A constructive alternative to punishment is to avoid anger and hurt feelings. Maintain appreciation in the relationship. Offer her chances to help and seek support and help in identifying positives about your child. Most of all, don't give up. You want to change her belief from no one really likes me to "I matter". She needs to feel significant and valuable. The positive goal to replace revenge is CONTRIBUTION.
If your daughter is seeking revenge, you may find it helpful to work with a professional as a child seeking revenge is very discouraged. I did not go into length about strategies for undue attention and power struggles as the situation sounds most like revenge. This information is based on the work of Rodulf Dreikurs who wrote Children: The Challenge.
In conclusion, I strongly encourage you to focus on building your relationship with her and helping her to build her relationship with her sister as well as with friends. Perhaps she can teach her sister something? Help you with something? I'm sure you will come up with some great ideas. It sounds like she really needs reassurance of her place in the family, that she belongs, is competent and matters.
Best wishes!