Almost all buses in Edinburgh are run by Lothian. Confirmation of that �2.30 price is here:
http://www.lothianbuses.co.uk/tickets/dayTicke ts.asp
(NB: Lothian Buses operate an 'exact fares' system. The drivers can't give change, so you must have the right money to hand before boarding).
Lothian buses run various tour buses. The �9 fare refers to the 'City Sightseeing' tour, which basically goes around the Old Town area:
http://www.lothianbuses.com/edinburghbustours/ Sightseeing/interactive_map/highlights.html
Most of the places along the route are within walking distance of each other. (It's about 1.5 miles from one end of Old Town to the other).
You can certainly get around all of the main tourist attractions without buying the �9 ticket. Either walk or buy a �2.30 day ticket. For a map of bus services, go here:
http://www.lothianbuses.co.uk/howto/findRoute. asp
(Click on the City Centre to bring up a detailed map for that area).
Chris