Yeah, if I remember right, their "connect to the internet" mode was a special dial string at the end of the dialup number that selected between normal Compuserve access, and Internet access. If you used Compuserve to get on the Internet, you were unable to access the normal Compuserve services through the Compuserve app! Not a great way to integrate. The only main integration point I remember was that you could use your Compuserve login ID as a full Internet email address.

(This was all back in the 90s when I was a teenager of course, so I may have gotten the details wrong, or just didn't know what I was doing when I was using Compuserve's Internet connectivity.)