It sounds to me, based on the replies, that is may be someone who is very ambivalent about leasing (and may be worried about the soundness of her horse) and is bending over backwards to protect herself, expecting the worst. However, given it’s an on-property half-lease, some of this sounds a little bit paranoid. I’d be more sympathetic to the owner if this was off-property and a full lease.
I leased horses for many years, and if there isn’t trust between the leaser, person leasing, and also the trainer(s) involved, it will not end well. (I know some people who lease horse do horrible things, so please don’t think I’m criticizing all owners). But it doesn’t sound like the owner is going into this with a sense of trust that you will do right by her horse and her tack. It’s fine to ask for $$$ regarding vet expenses in a contract, but there should be transparency about a ballpark figure about what the horse has cost in the past re: vet care. The image you’ve painted is someone who is looking for you to do something wrong, which feels awful.
What are stipulations in the contract about how often/where/when/ what you can do with the horse?