Well, from a moms point of view I can tell you that it is hard to let a child move on. We always want to see you as our babies. Every time my husband gets upset about his mom 'nosing' into his life...I remind him that she remembers what his dirty diapers smell like...were just a different breed. That being said, her being non-horsey makes things difficult.
Since you are going to be an intern/working student I would explain it this way. The horses that you will be riding will be students ALSO. The trainer can only ride one horse at a time. Its easier for them to crank out winners if they have good, competent riders (such as yourself) working under said trainers. You learn how to train and the horse is being trained by you and the trainer TOGETHER. Everyone benefits and no one loses. Explain to her that hiring less qualified trainers with ideas of their own doesnt benefit the stable. Im sure they have their own training method and if they are successful enough to be bringing on working students, it must be a good one. It is cheaper for them in the long run to pay for housing (which probably doesnt cost them much as I would imagine it is on the stable property) than it would be to pay x amount of trainers x amount of dollars for a various training styles. Your pay back is the same as being in college. You get to put them down as a reference and possibly get hired on after your internship.
In college, you would have a teacher standing in class teaching 30 people one concept. At the end of the day, you would have to pass tests and get graded. You will see this in your internship also. If you arent working out, you will hear about it and you wont stay long. The difference is you will be getting hands on education in a field where a degree helps but experience can really sway employers.
Instead of a diploma, you are working for a good reference. The difference is, you wont have all those pesky student loans at the end. Making money riding and caring for horses can be difficult and is usually left for those who are very talented riders. Explain that you feel that you have a natural talent for this and that this opportunity will just help you bring your talent to the next level.
As far as the car insurance goes, tell your mom that if you can't pay your car insurance, you will have to leave your car at home. You will catch rides to the grocery and you will probably be too busy riding to do that. Without a car, you wont be able to come home to visit. Its not nice to manipulate but...sometimes parents just need the motivation to help you chase dreams.
My own mom let me persue something similar to what you are doing allllll the way in California. I had a blast. After a year, I came home because my internship was done and my mom was sick. She never really wanted me to go but after she saw that it did make me happy, she was glad to help. Believe me, she wants you to succeed. She's just not sure...help her!
Good luck with everything! If its meant to be, that morgan will show up magically sound tomorrow and you can get your ride in. If not, you will find a way.