Yup-agree. Any trainer worth their salt does a contract as far as what the payment will be, expectations, etc. I know that here in the Mid atlantic, where things are pretty expensive, I paid $800/mo for training last winter. That included board, riding 4-5 days a week BY A TRAINER, and they helped me for free when I wanted to go ride. And-this was a top ranked reining SHOW barn! It was all in writing, no doubt what the expectations were on either side.
I would also bring the horse home. I would suggest that if you decide to do this on your own you do lots of watching of good training videos, and I personally would start in an area like a round pen, which IMO, makes it less likely for you to lose control. I would start very gently, from the beginning and make your own assessment of what your horse knows and doesn't. Take nothing for granted-treat him like he knows nothing, and be pleasantly surprised if he does. But-don't waste another dime on this woman.