I want to have a dedicated Proxy Server for my local network. What Ubuntu type, for example Ubuntu Server or Ubuntu Desktop should be used for this specific person. please help with some details and information about these distribution
It doesn't matter. The difference between Ubuntu server and Ubuntu desktop is the default package selection. You can run common proxy software, like squid, on both.
On the other hand - having a graphical environment on a dedicated proxy server does not matter either. You have to configure it using text files, and you don't need the graphical interface for anything. So the conventional setup would be a headless box with ssh login.