I agree on the first mowing being a per hour charge. The first mowing will be the slowest due to getting use to the plot. There can be wet areas, stumps, ground hog holes, old fallen tree parts, and trash/junk. I do not have many customers complain about charging per hour. In fact most of my new customers will ask me "how much do you charge AN HOUR to mow". So we go the per hour route. Its easier on me and I get paid for being more cautious on the first mowing. I tell them first mowing is slower and why, if they still want a finished price, they get it, but I guarantee it will be higher then charging per hour. If they want a per acre, I do that too. I can tell by looking pretty good and by my mowing pace, what gear and rpms how I am doing acres per hour.
As for damages, I carry insurance. Both a general liability and commercial inland marine policies. General liability fixes damage on customers property, commercial inland marine fixes damage to my equipment. I have not had to use either yet.
I will usually ask customer to meet on site for a go over, what they want mowed, property lines, what the area was used for and how long since it was last mowed, any known obstacles and I list them to help jog their memory, and then I will walk area a bit looking for hazzards. I have been around this type of work long enough that I can pin point the high risk areas. Other risks, I keep an eye out for when mowing.
I have hits rocks, tree parts, old stumps, a well casing, many ground hog holes, been stuck in mud, bees nests, glass beer bottles,property stakes, and old boards. Biggest thing i fear is a tire puncture. I have been lucky and not had a flat tire yet. Very little if any damage from what I mentioned above.
In lots of ways I feel a lot of contractors will bid a job at the highest rate they can get from a customer, then go balls to the wall to try to maximize their profit. They have this "make a ton of money" idea in their head. This leads to not being very alert when mowing and greatly increases the risk of damaging their equipment. I see it a lot actually. They think because they can mow faster, such as higher ground speed, faster gear jamming for forward/reverse, etc. Which I all harder on the equipment. Their equipment is in the shop much more then mine. So I bet you I make as much as they do with all the time actually involved.