I believe that there is a misconception about how IPs work here. When you say 127.0.0.1, you are talking about a very special IP that all NICs associate with the loopback interface. Basically, sending something to 127.0.0.1 is like sending them to an internal mirror which reflects everything back to your computer. Obviously, any other computer can't access your computer from by using the 127.0.0.1 IP, since every computer "thinks" that themselves are 127.0.0.1, and no other. Therefore, your cousins computer will reach himself by accessing the 127.0.0.1, and not you.
The range of 192.168.x.x are private IPs. Only computers in the context of your internal subnet ( with IPs in the range of 192.168.0.0 to 192.168.0.255, considering a 24 bit subnet mask ) will be able to access it.
Since, in fact, you were able to access yourself at that IP, every other machine in the subnet (for example, your smartphone connected to WiFi) should be able to access your Joomla application.
Does this make things any clearer for you? Where you trying to access 192.168.0.20 from outside?
Also, remember that, since the range of 192.168.x.x is private, it does not matter that another private network has the same IPs.
If, after this, things are not clearer for you, please don't hesitate to write us back and we'll try to give you a better solution.