I'm not getting the help I need

Your reasoning of 'making the 1st appointment for appearances' may be closer to the truth then you realise. If you go to see a doctor and he thinks you may be in danger either from yourself or other people (especially family members) he may call social services and your parents will be under scrutiny.
Your mothers' excuse of 'not finding a doctor she likes' is irrelevant. It's not about her; it's about you and she needs to realise this, quickly.
You must tell your youth director that you haven't been to see a doctor yet and your appointment was cancelled and nobody has followed it up. Ask him if there is anyway he can make the appointment for you, take you to the appointment without your parent's knoweldge until you feel strong enough to confront them. Or if that doesn't appeal to you, ask him for phone numbers for a doctor he can recomend and phone them yourself. Explain the situation to the doctor and ask for an appointment. Due to patient confidentiality the doctor can not tell your parents anything.

You must find a way to tell your parents eventually, either through a letter or face to face. Take a friend with you for moral support, sit them both down and explain that if you yourself can realise you need help to cure your depression and self-harming, then so should they. Tell them it doesn't matter about making appearances or what other people may think, all that matters is you being able to feel normal and supported rather then being thrown against a brick mural again and again.

Well, online FREE therapists are hard to find. But often (or mais than often) doctors or other people that aren´t close can rarley hep you as good as a friend or family member. You can talk to a good friend about this. Or a sibling. Like why you feel depressed. If you don´t know why, maybe it´s self-loathing, for some reasons you can or cannot change. The first thing you should do, (and if you don´t want to, you don´t even have to talk to someone else) is to find the probelm then look for ways to fix it. If you can´t get rid of the probelm yourself, you should get help from someone like a friend or adult. (online friends are fine, actually :) Or you could message me if you want to. xD I really hope you can find a solution! xP
Good luck! ")

is there someone ur scared of or were u bullied welll look u should tell ur mom that u still fill hurt if she cant understand go to someone who will sometimes that happens if someone said something to u and u feel hurt or somethings in ur life r messed up my conselhos is u tell ur mom that its not happening to her its happening to u so if she keeps sayin i dont like that doctor u must have a real long talk with ur mom to let her kno that u need to pick the doctor u want i kno she is ur mom but if u have self harm u need mom to understand and another thing if u have something to say dont bottle it upn inside llet it out