Things to ask about potential lease horse/contract?

I sold my mare last fall and am now looking for a horse to part lease. Obviously, I haven’t leased a horse in many years; even then the ones I’ve done in my younger years were pretty casual, handshake type agreements.

I’m planning to be very picky on the horse (I am not interested in dealing with vices or greenies anymore!). I want to make sure I get what I’m looking for, and also protect myself from potential grief or drama with the horse’s owner.

Is there anything I should be asking the owner or looking for in a contract?

Some of the basic terms that should be contained in the contract: identifies the horse, fee, what is included (who pays for shoes, grain, vet, whether you use the owner’s tack), how often you can ride (are there set days?), restrictions on type of use, whether you can take the horse off-site, length of lease (set term or month-to-month, how it can be cancelled), and waiver of liability.