Ummm, there's no such thing as needing to wait until you're an adult to get a hair cut! It's not a medical procedure. It's not a drug.
If you're not ready to come out, don't force it just because of a hair cut. If you want short hair, make your own appointment and just do it. Remember that parents have absolutely no parental rights over the length of your hair. I started getting my hair cut myself when I was 10. My mum always hated my spikey hair but she just had to deal with it.
My personal opinion: cancel the appointment (easy as a simple phone call), make your own appointment either there or wherever you like and get it cut how YOU want. Then use the confidence you gain from short hair as the path to coming out. The more you become the real you, the more confident you'll be in yourself and the more independence you'll both feel and project.