No, just setting a proxy within your browser not make you anonymous.
Your browser and flash is likely to have cookies that will identify you.
If you wanted to be truly anonymous you would probably need to fire up a new browser from a livecd or in a VM, set the proxy, and then browse. To maintain your anonymity you must not login to any site that has your name.
You will probably also need to make sure the browser/OS you are using does not make DNS requests to a local DNS server. If you do not configure the proxy correctly your machine make make DNS requests for the sites you are visiting to the local network.
If the proxy you are using is not using SSL for transport your ISP or someone on the local network will still be able to see what you are doing. Even if they are using SSL your ISP will know you are contacting that proxy.
If the an evil person cannot get the logs from the proxy machine to get the logs. if they are able to monitor the incoming and outgoing traffic they still may able to identify you just by looking at the flows of data.
Something like TOR does the best job, but even with it you must be very careful that you set it up properly and that during your session you never reveal any information that would unmask you.