Good for you for getting a plan. The thing is though, speaking from a parents perspective, kids are great at telling you what they will do, but so much better at not following through. If I allow my child to have an animal, it is ultimately my responsibility. A horse is a huge responsibility.
What you need to do is start showing your mother what you will do. Get yourself a job and open that savings account. Put every penny you can in it. If you have any pets at home, start taking responsibility for them. Clean up after them, exercise them and take responsibility. If you don't already take lessons, then start them. Volunteer at a local barn to help. Clean out some stalls, water, brush,etc.
Speaking from a parents perspective, don't tell me what you are going to do. Do it. Then we will talk.
My own daughter begged for a horse for a long time. It wasn't until she was taking riding lessons, cleaning stalls, scrubbing water buckets and working that I took her seriously. She now owns her own horse.