Make sure to read the policy and keep a copy with your lawyer too.
Insurance...folks who take tons of your money, and then when you have an accident use your money to hire legal goons, to sic on you...in order to avoid having to pay up.
A good policy will cover theft, personal injury, and collision, if you can get them to insure a moped, best bet on that is go with the insurance you have for car/home/etc and get them to "bundle" the policy.
Reason is, that's usually cheaper, and since you're dealing with people you know, and you can perhaps hold responsible for any chicanery, there's a lot less inclination to stick it to you.
But mark my words, don't ever, ever, trust them...know how to respond in force when they refuse to meet their end of the agreement, and don't be afraid to do so.