Final answer:
The firewall type that initiates connections on behalf of a client is a proxy server, specifically operating at the application level.
Step-by-step explanation:
The type of firewall that initiates a new connection on behalf of the client and presents its own IP to the server is known as a proxy server which operates at the application level. This kind of firewall evaluates outgoing traffic to a server by establishing its own connection to the server. The client connects to the proxy, and then the proxy makes a separate connection to the intended server, thereby isolating the client from direct exposure to the server.