I've purchased several horses on payments. Usually you discuss payments/down payment then the seller writes up a contract. If the contract is agreeable to both parties, you both sign and keep copies. If you're purchasing a horse on payments, it is absolutely critical that you keep insurance on the horse so that if something happens (colic, accident) and the horse dies, nobody's out any money. Also in the contract needs to be what happens to the money you've paid if for some reason you can't continue making payments. Very few people will allow a horse to leave their property without being paid in full. That's really best for both parties, IMO. Also you usually have to pay board and cover any vet costs associated with the horse including wormer and hoof care while you're making payments. Essentially, the horse becomes your responsibility as soon as you make the first payment.

Before you enter into any kind of agreement, you need to make sure you can make the payments and still be able to pay for emergency vet expenses. If you can't make the payment arrangements she requires, tell her that. If she's not willing to work with you then move on. You don't need to endanger you and your family's wellbeing because of a horse who already has a home. Just the fact that she needs to sell this horse to be able to afford feed is a huge redflag anyway. That tells me that money is her first priority because she doesn't have any. Just by her changing the terms of the sale makes me think she does have another buyer and is trying to force you to give up.

After all that, it just smells fishy to me. I know you care about *this* horse but let it go. Go spend your money on the well broke gelding you were talking about. I think it would be money more well spent.