Caching proxy server
Anonymizing proxy server
Transparent and non-transparent proxy server
The term "forced proxy" is ambiguous. It means both "intercepting proxy" (because it filters all traffic on the only available gateway to the Internet) and its exact opposite, "non-intercepting proxy" (because the user is forced to configure a proxy in order to access the Internet).
Forced proxy operation is sometimes necessary due to issues with the interception of TCP connections and HTTP. For instance interception of HTTP requests can affect the usability of a proxy cache, and can greatly affect certain authentication mechanisms. This is primarily because the client thinks it is talking to a server, and so request headers required by a proxy are unable to be distinguished from headers that may be required by an upstream server (esp authorization headers). Also the HTTP specification prohibits caching of responses where the request contained an authorization header.