You're right.
However, pinging the default gateway is pretty much the same thing as the first ping test listed in your post. 192.168.0.1 is a common default gateway address for SOHO routers. If you ping the default gateway and get a reply, it means all your network hardware is working, from the NIC, to the cable, to the router. You wouldn't have to ping 127.0.0.1 if you got a reply from the default gateway ping, since tcp/ip software is necessary to get a reply back from a default gateway ping. TCP/IP is assumed to be working if you can ping default gateway successfully.