It depends on the club, however as a general rule, you usually pay block time, i,e from when you first taxi, to shutdown. So, in your example, you would pay a total of roughly 2 hours.
Same at my club. There are some variations, depending to an extent to what equipment is installed in the plane. Some clubs use the honor system, requiring you to accurately write down either engine start/stop, or take-off/landing times, others use the tacho or a hobbs meter. (This leads to endless discussions at club meetings...) But at the end of the day, you normally only pay for flying time and time associated with the flying (taxiing, runup), not for time spending parked.
Most club/rental places offer a "wet" hourly rate, meaning: including fuel. The aircraft comes with a fuel credit card which you can use to uplift fuel at most places. And if for some reason you've got to pay yourself for fuel away from home base, they will have a reimbursement system/agreement to pay you back. If you plan to do serious touring, check this system out, because phoning ahead and looking for the best price can be very advantageous. Also, if you're going to cross the channel, find out who's responsible for claiming back excise duty.
Some groups use a "dry" hourly rate instead, meaning: excluding fuel. I don't know how this works exactly, having never experienced this, but it makes people more aware of fuel consumption and fuel management, so it encourages proper leaning and shopping around for best fuel prices.
Some clubs impose restrictions on minimum number of hours if taking the aircraft for extended periods. For example, if you hired an aircraft for a weekend, you may be charged for say 5 hours flying whether or not you actually fly for those hours. Obviously if you fly more than the minumum you pay for the total hours.
Depending on how well you know the people, how often you rent their planes for just an hour without any downtime and such, this is typically negotiable. But negotiate beforehand, not afterwards.
Also, if you take a plane away for a longer period of time, some of the responsibility for maintaining the plane is also assumed to be yours. This is not like rental cars where you have places all around the country that can help you. So take tie-downs and some oil with you. Perhaps a sponge, soap and a bucket. Make sure you have a list of phone numbers of people/places to contact in case of problems or questions. If you can't fly your plan due to weather or aircraft problems, contact the club as soon as possible so that they can shift other renters to other aircraft. Etc. etc.
Oh, and if you plan to go away for a longer period of time (say 5-10 flying hours at least), check with the rental place a few days to a week before, to see whether there's enough time left until the next 50/100 hour check. If you do this well enough in advance, they can schedule the check a few flying hours early so that you don't get stuck in the middle of nowhere with a plane that ran out of hours.