What isn't working as expected, is the "limit_web_ip." What I am expecting is that when the user creates a new site, they will only be able to use the IP I set in this field, assuming it is one of the server's configured IP addresses. Am I correct?
Youre wrong, the Limit you are searching for is set in the server_ip database table and not in the client table. See ispconfig interface: System > Server IP, there you assign the IP addresses to a client.