Since the horse is staying in the owners possession until she is paid off, she is responsible for any injuries to the animal and its vet care until official ownership is transferred to you.
You need someone to help you write-this up, I would not be comfortable sending money without a contract either.
Usually there is a non-refundable deposit that the owner would keep if you are unable to make the payments and cancel the sale of the horse. For example, if the deposit is 500 dollars and you give her 1200, but something happens and you have to cancel the sale, you should get 700 back.
I would also put that any serious injury that leads to soundness issues or death will result in full refund of any payments made (Including deposits.)
Make sure you both agree to the terms before you send any money.
Some big-breeding stables have "Foal Contracts" that are listed on their site, since many people put deposits on weanlings and don't get them until weaned (and paid.) You could try looking around for that and tailoring one to fit you and the current owner.
EQUUS KEEPUS BROKUS