After reading your other thread, I must admitt I closed my eyes and took a deep breath when I came across this one.
You are 13 years old, you are NOT experienced, riding a couple of lessons horses is not experienced, experienced is riding many different horses of different abilities, educating them, re-training them, staying on their bronc sessions etc. I have worked for 2 dealers in the past, riding up to 5 'turn over' horses a day (often picked up from the doggers because they were notorious buckers or similar), ottb's, spelling racehorses, breakers, and then the edicated horses, I've started with 3 year old and taken them up the levels to elementary/medium dressage and sold them on. And I do not, by any means, consider myself an experienced rider.
Even now, looking for a horse, I am looking for something above all else, quiet, as well as being able to move and have ability for the higher levels of dressage, I do not, by any means, consider myself an 'experienced' rider enough so to take on a rescue case or similar.
The dream of taking on a resuce is great, you're going to take it on, feed it up, develop an amazing unbreakable bond with it because you saved it's life, then it will let you and only you ride it, doing absolutely everything and winning every competition you enter. Of course that's the dream, but it's far from reality.
Rescue's may be cheap to buy outright. But the financial outlay to maintain them is enormous!!!! First you'll need the vet bills, and generally, with a horse so severely under weight there will be substantial medical issues needing to be addressed and treated, often taking months to fully recover. The feed will be a huge expense, PARTICUARLY if you get a tb, 90% of tb's are not good doers, and it takes a hell of a lot of food, lots of experimenting with different types of feeds, and a lot of knowledge of the mineral needs of a horse to keep it going. You need to know what vitamins/minerals are lacking from the diet, often done by a blood test (again, $$$$$$) and you need to feed accordingly. You can't just go stuffing a resuce full of as much food as it will eat, you will probably kill it. Conditioning takes time, months to years in fact.
Another issue, as someone addressed above, is that a horse maybe dead quiet when they are skinny and unwell, but once you start feeding them and getting them fit, they can turn into explosive horror heads and you start to get an understanding of why they ended up where they did in the first place.
I really do think that a lease would be your best option at the moment. I know it sounds all nice and fairy tale to be 'saving' a horse, but what about your own life? If you're not experienced enough to manage a rescue case, it's very likely that you will end up putting yourself off horses because you will lose your confidence when the horse starts to find his feel and feed good again. And it will possibly end up exactly where he came from.
A lease would be the logical, intelligent way to approach your entry into horse ownership. You don't need to pay for the horse outright and you may not even need to pay for the full upkeep of the horse either depending on the contract. And if you don't get on with the horse, you don't need to go through the stress of selling it, you can just give it back.
Think with your head, not your heart. That's the way to go when dealing with horses.